LANGE  LIBRARY  OF  EDUCATION 

UNIVERSITY  OF  CALIFORNIA 

BERKELEY.  CALIFORNIA 


UNIVER- 

BERKELEY. . 


A  study  of  eliminations,  inclusions,  and  social  and  business  requirements 
of  arithmetic 

By 

Katherine  Soiers 

B.L.  1914 


Submitted  in  partial  satisfaction  of  the  requirements  for  the  degree  of 


MASTER  OF  ARTS 

in 

Education 

in  the 

GRADUATE  DIVISION 

of  the 

UNIVERSITY  OF  CALIFORNIA 


Approved  Ej  AD 

Instructor  in  Charge 


Deposited  in  the  University  Library 

Date  Librarian 


: 


EDUCATION  OEPT. 


TABLE      OF      CONTENTS 


PART      ONE 


PAGE 
INTRODUCTION    1-      2 

THE   PROBLEM 

THE  MATERIALS 

THE  METHOD  OF  APPROACH 
CHAPTER   I  —  THE  NEED  FOR  SCIENTIFIC  INVESTIGATION       3-  14 

IN  EDUCATI 0* 

CHAPTER     II--   AIMS    IN  ARITHMETIC 15-19 

TRADITIONAL 

PRACTICAL 
CHAPTER  III—  ELIMINATIONS  IN  ARITHMETIC   WHICH    HAVE  20-   49 

BEEN   SUGGESTED  OR  MADE 

CHAPTER      IV—    MINIMUM  ESSENTIALS   IN  ARITHMETIC 50-    63 


6*ttfi23 


PART  TWO 


THE  INVESTIGATION. 

CHAPTER   I—  SOURCES  AND  METHODS  OF  COLLECTING 

THE  D  SEA 

QUESTIONNAIRE  TO  THE  PUBLIC 

QUESTIONNAIRE  TO  PARENTS 

QUESTIONNAIRE  TO  BUSINESS  FIRMS 
CHAPTER  II—  QUESTIONNAIRE  I 

TABLES 

CHARTS 

JUDGMENT 
CHAPTER  III—  QUESTIONNAIRE  II 

TABLES 

CHART 

JUDGMENT 
CHAPTER   IV—  QUESTIONNAIRE  III 

TABLES . 

CHARTS 

JUDGMENT 

CHAPTER   V  —  SUMMARY  

I  NATION  OF  TOPICS 

ESSENTIALS  IN  SUBJECT  MATTER 

CONCLUSION 


PAGE 
64-  68 


69-  85 


86-  91 


92-  96 


97-100 


PAGE 
A  SUGGESTED  COURSE  OF  STUDY  IN  NUMBERING  IN  THE  GRADES . .101-109 

ESSENTIALS  IN  PROBLEMS 

BIBLIOGRAPHY 110-  H8 

APPENDIX    

CRITICISM  AND   SUGGESTIVE  CHANGES 

FOR  THGRNDIKS'S  ARITHMETICS. 


-1- 


INTRODUCTION. 

The  thesis  I  maintain  is  this: 

Tho  kind  and  amount  of  arithmetic  taught  in  tho  elemen- 
tary school  and  the  method  of  its  teaching  rest  upon  the  demands 
of  the  business  and  the  social  world. 

In  establishing  this  thesis  I  shall  approach  tho  prob- 
lem in  tv;o  wr.ys:  first,  by  examining  the  accessible  writings  of 
recognized  educators  and  by  collecting,  collating  and  organizing 
the  records  of  what  has  been  done  by  investigators  up  to  this 
time  in  meeting  the  needs  of  society,  in  setting  up  aims,  in 
determining  essentials,  in  eliminating  useless  suV ject-mattor, 
in  proposing  methods  and  means  of  procedure,  and  in  compiling 
practical  courses  of  study;  second,  by  first  hand  investigation 
and  study  of  the  requirements  of  society,  I  shall  endeavor  to 
search  out  tho  minimum  essentials  of  arithmetic. 

U-ing  the  data,  records  and  judgments  collected  under 
the  first  section  of  this  study  as  a  basis  for  the  comparison 
and  testing  of  the  later  findings,  I  ho.->e  to  determine  in  what 
ways  and  to  what  extent  thfl  subject-matter  may  be  further  sim- 
plified, in  how  far  the  bulk  of  material  may  be  reduced,  and  how 
great  a  decroase  in  tho  time  devoted  to  the  teaching  of  arithme- 
tic may  be  made. 

By  inquiring  into  current  business  practices,  I  Shi  11 


-2- 


raako  an  effort  to  of  ?er  a  harmonization  of  school  methods  and 
vrorld  methods  of  procedure,  and  shall  succe3t  certain  ii. elusions 
demanded  by  social  needs. 

Finally,  I  shall  include  a  course  of  study  in  arith- 
metic founded  upon  the  facts  and  truths  established. 


CHAPTER  I 
THE  HEED  FOR  SCIENTIFIC  INVESTIGATIONS  IN  EDUCATION. 

,vrhere  is  no  form  of  knowledge  go  complete  and  final 
that  it  cannot  be  improved,  no  single  human  art  so  perfect  that 
it  cannot  be  made  better,  no  form  of  human  endeavor  that  does 
not  call  for  further  effort.  For  this  philosophy,  life  Is  a  per- 
fecting, not  an  arriving  at  perfections,  an  1  the  joy  is  in  the 
process,  not  in  reaching  and  remaining  at  the  goali* 

The  best  teaching  of  truth  is  that  which  seeks  to  define 
not  truth  itself  as  something  already  complete,  but  the  nature  of 
the  effort  required  to  search  it  out. 

The  birth  and  growth  of  the  idea  of  investigations,  edu- 
cational and  social  surveys,  and  general  stock-taking  for  every 
sort  of  institution  lias  been  within  the  lifetime  of  the  present 
•eneration.  The  idea  did  not  spring  into  existence  full-formed, 
out  is  developing  gradually,  and  is  marked  along  the  way  by  revo- 
lutionary principles.   This  movement  in  the  elementary  schools  is 
of  special  significance,  for  elementary  school  education  has  ful- 
ly entered  the  experimental  period.   Rapid  changes  are  being 
made  in  school  subjects  through  the  influence  of  modern  scien- 
tific research.   The  initial  need  is  to  know  whither  our  pre:; 
school  system  is  taking  us,  and  if  by  critical  examination  it  bo 
determined  that  our  direction  is  wrong,  wo         '  out  e  uca- 

1.   Moore,  Ernest  Carroll:   What  la  Education?  1914,  page  141. 


-4- 


tional  scouts  to  spy  out  direct  roads  to  a  pioneer  land  where 
we  may  find  the  truth. 

V/hat  we  have  in  education  is  an  inheritance  from  our 
forefathers.   It  La  the  product  of  indiscriminate  borrowing, 
modified  as  need  forced  modifications  upon  us.  Prank  experimen- 
tation is  imperative.   Our  feneration  must  shake  off  tradition 
and  organize  a  sound  scheme  for  educational  advancement.  Changes 
must  not  be  made  according  to  the  whims  of  individuals  —  the 
"cut-and-paste"  method  by  which  the  average  school  superintendent 
compiles  :i3  course  of  stiidy  is  tragically  common  --  nor  must 
these  changes  be  made  by  local  committee?  transient  in  character. 
Only  after  theories  have  been  proved  by  investigation  according 
to  scientific  method,  and  have  been  successfully  tested  and  tried 
in  the  school  of  practice,  may  they  be  pronounced  true  and  good. 

Many  problems  pertaining  to  the  construction  of  course! 
of  study  already  have  been  solved,  and  an  encouraging  body  of 
scientific  data  has  been  secured.  Many  foolish  things,  no  doubt, 
have  been  done  under  the  shield  of  experimentation,  but  not  ad- 
hered to  by  the  conscientious  worker  in  the  field.   He  may  make 
mistakes,  but  he  discovers  and  corrects  them  when  he  te.:t3  his 
theories  by  practice.  The  markets  arc  stacked   it.h  books,  and 
eager  teachers,  ritr  more  enthusiasm  than  wisdo  ,       ooradic 
trial  of  this  and  that  new  idea.   Opinion  serves  for  informal   n, 
and  the  disastrous  results  often  bring  disrepute  upon  a  system 


of  attack  which  is  striving  zealously  to  advance  education* 

risons  must  be  made  between  the  conclusions  reached  in  orig- 
inal  research,  and  the  results  obtained  by  previous  investigators 
alone  a  Ilka  line  before  Judgments  can  be  made.  And  then*  that 
no  reforms  nay  be  offered  which  have  not  stood  the  toot  of  our 
own  trying-ground,  every  fact  advanced  must  stand  the  test  of  ap- 
plication; for  the  only  effective  way  to  determine  whether  the 
conclusions  arrived  at  by  experimental  study  are  feasible  is  to 
see  that  they  work  out  well  in  practice  under  the  conditions  of 
the  classroom.  There  must  be  no  gap  between  c ducat :    L    ori- 
montation  and  school  practice.   Fortunately,  in  the  school  world, 
there  is  a  general  awakening  to  a  sense  of  values  baaed  upon  sci- 
entific analysis.  By  discovery,  surveys,  facing  and  solving 
problems  which  individuals  and  cownamitlea  have  to  meet,  we  are 
led  away  ft  .      ogaatical,  the  formal,  and  the  untrue. 

Our  investigations  and  experimentations  must  bo  based 
upon  sound  educational  aims.   To  make  the  discovery  of  anything 
worth  while,  we  must  know  what  human  purpose  is  to  be  served  by 
its  discovery.  7,'hat  is  our  problem?  What  is  our  purpose?  What 
are  we  trying  to  do?  What  is  our  Aim?  Ws.  must  have  no  groping. 
As  General  Foch  said,  "There  13  but  one  manner  of  considering 
every  question,  that  is  the  objective  manner."  V/hat  is  our  Objec- 
tive? 

Aims  in  education  established  by  asters  seen  to  vary: 
Knowledge  getting,  self-realization,  social  efficiency,  culture, 


■6- 


achicvenent,  adjustment,  learning  to  use  t.ho  tools  which  the 
race  has  found  L  sable,  duty,  growth,  utility,  complete 

living,  discipline,  —  ho.;  each  aim  calls  to  mind  the  ::ane  of 
its  advocate,  name  and  aim  closely  locked.  —  Christ* s,  "I  an 
cone  that  they  night  have  life,  and  that  thoy  ni         it  more 
abundantly,"  is  an  educational  aim.   These  ideals  establishc 
profound  thinners  seen  so  numerous  one  is  overwhelmed  v:hcn  con- 
fronted by  them.  Have  these  scholars  who  have  spent  their  years 
examining  education  arrived  at  diametrically  different  conclusions? 
Are  self-realization,  efficiency,  culture,  achievement,  complete 
living,  utility,  growth,  aims  separate  and  apart?  May  they  not 
be  aspects  of  one  great  requirement?  May  not  the  notif  running 
through  all  be,  "'.7e  serve  life"? 

Broadly  speaking,  Ml  nay  shift  advocated  educational 
ains  into  three  general  groups:   knowledge-get ti :.g,  discipline, 
use. 

Knowledge  getting,  when  knowledge  fop  ':.:ow- 

le  kg ■:■  is  the  aim,  is  an  old,  an  established  aim.   kith  this  the 
end  of  educati-n,  the  seeker  is  proscribed  facts  regardless  of 
the  usefulness  of  these  facts.   The  schools  are  knowledge 
sarles  '    -acts  aro  parcelled  out  by  those  who  neks  knowledge 
bestowing  a  business.   The  admonition,  "Hold  fast  all  I  give  you," 
accompanies  the  largess,  and  a  strict  accounting  of  the  accumu- 
lated information  is  required.   "Remember  and  you  are  educated," 
is  the  motto.  Arithmetic  facts,  geography  fasts,  spelling  facts, 


-7- 


history  facto,  literature,  art  and  1        f-icts,  thousands  and 
tons  of  thousands  of  thon  compounded  aro  forced  upon  tho  trusting 
ones  who  coma  asking  for  that  which  will  cure  their  ignorance. 
Urs  to  the  past  tv/enty-five  years  teachers  universally  accepted 
the  decree  of  tradition,  and  made  little  effort  to  select  aaablC 
from  useless  facts.   They,  with  the  iid  of  a  few  standardized 
texts,  prodigally  passed  out  f-icts.   The  key-word  was ,  "Know  for 

Ice  of  knowing."   "Hoy/  much  do  you         at,  "What  can 
you  do  with  what  you  know?"  la  the  shibboleth  of  those  who  lay 

.preme  emphasis  upon  the  acquisition  of  knowledge  ac  the  end 
of  education. 

Tho  modern  educational  world  is  striving  to  overcome 

1  2 

this  doctrir.e.  The  Ideals  of  such  leaders  as  Hellarry,  Cubberley, 

Dewey, °  Hall,  O'Shea,^  "loore,6  Thomdike  and  others,  aro  becom- 
ing directin  forces.   Dewey,  for  example,  utterly  opposes  infor- 
mation as  the  end  of  education.   He  deprecates  the  accumulation  of 
facts  for  recitation,  and  the  hoarding  of  nowledge.   "Knowledge 

Characteristic  V/o: 

1.  Ue Hurry,  Frank:   Elementary  School  Standards.  1013. 

2.  Cubberley,  Ellwood  P.:   Changing  Conceptions  of  Education.  1909, 

3.  Dewey,  John:   Democracy  and  Education.  191G 

4.  Hall,  .     Lays  Adolescence .  1904.  Educational        .  1911, 

5.  o'Chea,  Vincent <   Education  as  adjustment.  1903. 

Dynamic  Factors  in  Education.  190G 

6.  Hoore,  Ernest  C:  that  is  Education?  1914. 

oru.'cikc,  Edward  Lee:   Educational  Psychology.  1014. 


in  the  sense  of  information)  neans  the  vorkln  c  i  ltal,  the  in- 
dispensable resources,  of  further  inquiry;  of  finding  out,  or 
learning  more  things.   Frequently  it  is  treated  as  an  end  in  it- 
self, and  then  the  goal  becomes  to  heap  it  up  and  display  it  when 

I  for.   This  static,  cold-storage  ideal  of  knowledge  is  in- 
imical to  educative  development.1^"  It  is  particularly  mischievous, 
he  holds,  because  the  cramming  of  the  mind  with  non-perti 
facts  not  only  lets  occasions  for  t  .      go  unused,  but  swamps 
thinki   . 

;.."oore,  beliovos  that  knowledge  simply  for  knowledge 
sake  is  impractical;  it  Is  neither  warranted  by  facts,  nor  by 

.)lcgic  1  tests.   The  purpose  of  the  investigator  is  to  ac- 
quire knowledge,  but  when  the  facto  arc  in  his  possession  they 
must  meet  the  question,  "Yftiat  are  they  worth?"   Since  we  arc  pri- 
marily concerned  with  living  r  .thor  than  kn         select  those 
things  which  constitute  vital  knowledge.  Knowledge  as  the  oowir 
to  act  excludes  all  useless  information. 

Teachers  and  admini;  trators  are  profiting  through  an 

.the 
examination  of  the  work  done  in  the  Francis  barker,  and/Horace 

llann  schools.   School"  arc  being  looked  upon  as  places  where 

new  ideas  may  bo  evolved;  and  new  discoveries  made  and  tosted. 

The  ideal,  knowledge  for  knowledge  s  kc,  vhlch 

1.  Dewey,  John:   Democracy  and         n,  page  105. 

2.  Moore,  Ernest  C:   What  is  Education':'  Cliaptcr  II. 


■9- 


played  so  prominent  a  part  in  determining  the  curricula  and  the 
methods  in  our  schools,  which  has  evaluated  studies  according  to 
their  ability  to  impart  information,  is  giving  place  to  that 
knowledge  which  arises  from  use  and  is  for  use. 

The  doctrine  of  general  education,  or  formal  discipline, 
is  a  second  heritage  from  the  past.  The  belief  that  training  re- 
ceived in  one  line  of  mental  activity  spreads  to  other  linos  of 
mental  activity  is  behind  much  of  what  we  do  \r\   elementary  scho>ls, 
in  high  schools,  and  in  colleges.  This  great  assumption  Ls  the  one 
upon  which  education  has  rested  for  many  centuries.  Studies,  ac- 
cording to  the  advocates  of  this  theory ,  have  magic  powers.  They 
discipline,  develop,  and  perfect  minds .  They  are  to  be  pursued 
not  because  they  :  ave  specific  values,  but  because  t:;ey  improve 
the  mental  facult5.es.  Observation,  emotions,  reason,  will,  memo- 
i  be  trained  in  wholesale  fashion.  It  is  the  schoolmaster's 
place  to  make  of  his  classroom  a  mental  gymnasium  where  his 
pupils'  mind:..,  ses  upon  the  apparatus  furnished, 

may  be  made  more  agile,  keener,  strongor.  The  more  difficult 
the  apoaratus  is,  the  finer,  abler,  and  more  nearly  perfect  the 
resulting  mind  will  be.  Greek,  Latin  and  higher  mathematics 
Lave  ranke  1  gymnasium.  A  Marathon 

throu  science,  a  strenuous  wrestling  bout  with 

gilistic  encounter  with  algebra,  daily  try.: 
out  tournaments  with  3ome  hundred  or  so  of  the  400,000  words  of 
our  English  speech  are  supposed  to  Increase  mind  power.  The 


■10- 


gentler  calisthenics  of  poetry,  tart,  n  ic  will  lend  |  raco. 

Here,  too,  imperfect  minds  may  be  toncd-up,  burnished,  repaired. 
Tlie  different  types  of  minds  are  little  considered  and  the  pupils, 
in  groups,  pass  with  lock-step  regularity  from  exorcise  to  exer- 
cise throughout  tlie  day. 

If  we  turn  to  experimental  studies  made  upon  this  sub- 
ject, and  judge  by  the  writings  of  scholars,  in  the  past  twenty- 
five  years,  we  cannot  fail  to  recognize  the  error  of  this  doc- 
trine.  The  data  of  those0  whose  bias  is  frankly  favorable  to 
the  theory  seem  the  most  conclusive  against  it.  But  old  custom 
has  imbedded  it.  Perhaps  as  many  as  three -fourths  of  the  teach- 

f  to-day,  ind  undoubtedly  as  many  as  nine-tenths  of  the  edu- 
cated parents  adhere  to  it  as  fix1  &  ngh  faculty  psycholc  y 
had  never  been  questioned.  But  Schools  of  Educat"   a  I  research 
workers  ars  making  headway.   The  newer  I        s  freer  of  this 
pedantry  than  the  older  East.   This  confusion  which  began  in  that 
far-away  day  when  the  Sophists  taught  that  if  one  wanted  to  I 
physician  he  must  study  rhetoric,  and  if  he  wanted  to  be  a  general 
he  must  study  speech-making  must  be  completely  cleared  away. 

1.   Dewey,  John:   Democracy  and  Education. 

:iooro,  Ernest  Carroll:   What  is  Education'.  1914. 

•   Rugg,  II. 0.:   The  Experimental  Determination  of  Mental  disci- 
pline in  School  Studies,  1916 
"ooro ,  Charles  II.:   The  Inadequacy  of  '         against 

slplinary  Values.   School  and  Society.  toe ,29 ,1917. 
Coover,  John  Edgar!  Formal  Discipline  frc        indpoint 
of  Ba  ology.  101G. 


■11- 


hip  ile  says,  "The  problem  of  mental  discipline,  of  determining 
under  what  conditions,  by  what  methods,  and  to  stoat  extent  train- 
inj  in  a  given  line  of  mental  activity  extends  to  other  lines  of 
menial  activity  is  acknowledged  to  be  the  central  problem  of  edu- 
cational psychology."   Uoorc  says,  "Formal  discipline  is  the  cen- 
tral problem  of  educational  philosophy,  and  the  attitude  which  we 
who  teach  take  upon  this  problem  determines,  as  ..  , 

what  wo  put  into  courser;  of  study,  and  how  v/c  teach  that  which  we 
attempt  to  teach." 

This  doctrine,  undoubtedly  a  pmicious  :    alnfUl  error, 
will  ^o  the  way  of  worn out  superstitions  only  when  the  ancient 
idol  lias  been  completely  demolished  by  painstaki        rch. 

Use,  the  third  educational  aim,  we  hold  is  education's 
coal.   Our  real  reason  for  including  studies  in  our  curriculum 
ia  that  they  are  indispensable  helps  to  us  in  life.   Let  our 
words  to  our  classes  be,  "You  ar.  here  to  learn  to  do  certain 
things  which  the  race  has  found  that  it  cannot  live  without  C 
Every  lesson  lias  a  specific  ai.       ,/ou  are  first  to  see,  and 
then  if  possible  to  accompli ah.   The  question  for  you  at  all 
stages  is,  'Can  you  do  these  socially  necessary  things''"       , 

1.  Whip  lo,  '  .   .  :   Preface  to  Ru.   '  1  na- 

tion of        iaclpline  in 
'oore,  E.G.:  Address  to  Superintendents,  state  Normal  School. 

Feb.  9,  1918. 
3,  nooro,  B.C.i  'intendenti  ,     '        tate 

Normal.   Feb.  9,  1918. 


our  schoolroom  is  a  work  shop  where  children  learn  to  use  the 
tools  which  the  race  has  found,  indispensable.   It  is  of  i-imeasur- 
able  Importance  that  the  things  which  a  child  3ponds  his  ' 
upon  shall  he  of  immediate  use,  and  shall  servo  in  the  future  in 
such  a  way  that  he  shall  go  on  using  them  and  increasing  hll 
tery  of  them,  "..e  must  eicaraine  the  needs  of  the  adult  world  and 
fti   :,jz   and  girls  to  acquire  the  beginnings  of        ledge 
and  the  skills  deaanded  for  the  future. 

The  writer,  by  means  of  personal  Interview  ,    I  ivored 
Lscover  from  the  people      Ives  what  iv   e  In 

boys  and  girls  to  school.   "What  do  you  expect  the 
schools  to  do  for  your  child?"  was  the  essence  of  the  quest! 
asked  of  one  hundred  parents  in  many  different  Wi        life. 
College  professors,  alnlstere,  bankers,  physicians,  ranchers, 
plumbers,  firemen,  grocers,  shoemakers,  men  end  women  in  offices, 
shops  and  3tores,  truck  drivers,  the  scissors  grinder ,  the  gar- 
bage collector,  and  the  Chinese  green  grocer  were  among  those 
questioned.   The  reasons  were  varied;  indeed,  almost  as  numerous 
were  they  as  the  people  who  offered  them: 

"To  train  hie  nind." 

''To  learn  the  things  he  will  need  to  know  to  get  on  in 
life." 

"To  train  his  reasoning  powers*" 

"To  got  knowledge." 

"So  he  will  be   cultured." 


"To  bead)  him  to  live  wit  -,  and  to  pick  up 

a  bit  of  knowledge  is  he  goes  along." 
"To  teach  him  to  think  logically." 
"to  learn  to  speak  -  id  rrlte  nicely." 
"I  don't  knew.   I*ve  never  thought  about  it.   "very- 
body  rends  children  to  school.  They  have  to."  (Custom. ) 
"So  he  nay  help  tore,  d  write 

letters." 
"Ho  he  will  know  -ore  and  make  a  bet+    '•  '    than  1 

have  .r.ado." 
It  ^as  Interesting  to  note  that  U  ted 

parents  asked  for  knoi       Ilsclpline,  culture.  As  one  vent 

the  financial  and  social  scale,  the  demand  for  e  useful  edu- 
cation increased.   Products  of  colleges  and  schools  cling  to  the 
traditional  reasons  for  educating;  the  more  uneducated  classes 
j  practic-  1  reasons. 

Tabulated  the  results  stand: 

1.  Skills,  and  usable  knowledge  SO 

2,  General  education  —  discipline       27 

'-  idge  getting,  1G 

4.  Culture  (four  of  these  college  pro- 

fessors.) 8 

5.  Social  adjustment  7 
C.  Custom.  o 

Tot.-  1         100 


-14- 


Thesc  data  are  offered  ac  an  interesting  showing  of  the  public's 
reaction  to  a  mutable  question,  not  as  proof  of  anything;. 

If  we  accept  use  as  our  standard,  spendin     .   or- 
tional  fraction  of  each  day  in  the  company  of  reading,  writing, 
arithmetic,  geography,  history,  et  cetera,  will  not  do.  We  must 
select  the  studies  and  the  parte  of  the  studies  which  we  empha- 
size; there  must  be  a  purposive  accumulation  of  facts.   Our  sole 
aim  in  learning  anything  must  be  to  learn  to  use  it.   Plato  has 
phrased  it  "teaching  the  young  the  knowledge  which  they  will  after- 
wards require  for  their  arts." 

In  this  educational  turmoil  there  is  one  confusion  ee 
must  keep  elear  of.   '..hen  we  say  all  training  shall  be  specific, 
we  do  not  nean  that  education  shall  be  narrow  or  circumscril   , 
nor  that  subjects  shall  be  walled  off  into  compartments.   Instru- 
ments in  the  use  of  which  we  have  acquired  skill  must  be  constant- 
ly available:  our  whetted  axe  is  a  general  tool.   If  we  ar,  to 
substitute  the  direct  attack  for  the  superstitious  routine,  our 
first  step  naxst  be  to  gather  the  tools  which  will  contribute  to 
that  result. 


-15- 

CIIAPTER  II 

ttlE  IN  ARITHMETIC. 

Arithmetic  has  held  a  commanding  place  in  the  school 
curriculum  for  many  years  and  lias  developed  for  its  justification 
a  series  of  educational  aims,  broadly  spo  aking  those  may  be  sep- 
arated into  two  groups: 

1.  Traditional  Aims. 

2.  Practical  Aims, 

Traditional  aims  claim  for  arithmetic  all-encompassing 
virtues.   Scholarly  articles  have  been  written  to  show  its  disci- 
plinary T   cultural,  aesthetic .  and  ethical  values.  The  purely  dis- 
ciplinary basis  has  justified  the  inclusion  of  hundreds  of  social- 
ly unnecessary  tc  ics,  and  thousands  of  archaic  wad  artificial 
problems.   Claim  is  made  that  contact  with  absolute  truth  aids  in 
setting  up  higher  standards  in  all  kinds  of  work;  tliat  reason, 
judgment  and  concentration  are  trained;  tliat  careful  analysis  is 
valuable  in  that  it  leads  one,  in  considering  every  problen  in 
life,  to  exclude  the  non-essentials  and  hold  rigidly  to  a  definite 
line  of  argument,  by  setting  down  results  with  clearness,  and 
stating  theories  with  tersenec  ,  .    I  order  is  evolved.  The   cul- 
tural value  has  a  certain  literary  claim.   In  one's  dally  reading 
such  expressions  as  "a  changlxi  ratio,"  "tl  '   .  - 

trenes,"  "a  proportional  relationship"  etc.  do  arise,  but  to  jus- 
tify instruction  in  whole  to:>icc  for  the  sake  of  literary  expres- 


■16- 


context  is  absur".   An  aesthetic  value  is  claimed.   ;;ythm  and 
orderly  arrruigencnt  arc  supposed  to  inculcate  an  appreciation  of 

eautiful;  keen  pleasure,  oxalation  of  mind,  a  thrill  of  joy 
nay  accompany  a  student's  Q.E.D. ,  but  that  night  '.7ell  arise  from 
the  accomplishment  of  any  completed  v/ork.  A  careful  study  of 
arithmetic  is  claimed  to  promote  ethical  standards,  though  what 
there  is  in  the  study  to  make  one  treat  his  fellow-man  better, 
or  love  his  country  more  is  difficult  to  say. 

That  traditional  aims  dominate  courses  of  study,  and 

l,he  v/ork  mapped  out  is  largely  in  terms  of  traditional  sub- 
ject matter  is  evident  in  eight  out  of  ten  of  the  courses  in  use 
in  our  city  and  county  schools.   Certain  California         hools 
have  only  recently  awakened  to  the  fact  that  a  new  era  in  ari 
tic  has  dawned.  The  course  in  use  in  the  Training  School  of  the 
Los  Angeles  State  Normal  School  in  1917  contained  the  following 
statements: 

"The  cultural  and  ethical  phases  of  Arithmetic  must  be 
recognize  I  if  we  are  to  make  our  subject  of      .   educational 
value.  A  great  re  ponsiMllty  rests  upon  the  teachers  of  arithme- 
tic. There  is  const  nt  opportunity  for  encouraging  clear-cut, con- 
cise statem jnts  of  conditions.   Laxness  and  ccrelccsnes.-  in  speech 
beget  inaccuracy  of  thought)  and  a  disregard  for  logical  sequence. 
Poverty  of  mind,  and  a  patent  lack  of  understanding  of  data  involv- 
ed, are  evidenced  by  inaccuracy  of  statement.   CIc  r,   :.  jt, 
coherent  and  graraaatical  expression  is  of  vital  importance  in 


17- 


Ari  thine  tic. 

"Arithmetic,  indeed,  has  a  moral  worth.   It  gives  the 
child  an  opportunity  for  overcoming  obstacles;  it  develops  In  him 
a  broadminded,  self-sufficient  attitude  toward  difficulties.  It 
disciplines  him  in  persistency  in  effort  toward  accomplishment;  it 
serves,  in  a  degree,  as  a  check  to  flabby  inefficiency  of  thou  lit , 
and  loose  vagueness  of  conclusion.   It  makes  of  him  an  invest i  it- 
or,  a  questioner,  a  doubter,  a  scientist.  The  feelinf  of  certain- 
ty must  be  his  if  he  is  to  attain  the  joy  of  achievement,  and  the 
intellectual  delight  ?<rhich  comes  through  mastery. 

"Much  val\iable  information  is  rained  through  the  medium 
of  Arithmetic.  Affairs  of  the  business  vorld  —  customs,  forms, 
procedures, —  are  dealt  with  and  lastingly  impressed  through  prac- 
tical problems.   Hundreds  of  facts  imperative  to  an  understanding 
of  the  industrial,  commercial  and  social  world  are  constantly  being 
added  to  the  pupil's  knowledge-content  without,  in  any  sense,  dim- 
inishing his  opportunity  I        ng  arithmetical  proficiency." 

Dr.  Frederick  Burk  of  the  San  Francisco  Normal  School 
was  among  the  first  of  the  western  school-men  to  challenge  the  ac- 
cepted arithmetical  aims,  and  to  declare  for  a  revolution  In  sub- 
ject-matter. He  proclaimed,  characteristically,  that  there  was 
"something  radically  rotten  in  the  State  of  Arithmetic  in  elemen- 
tary schools,"  and  recommended  that  "the  podago^lcal  skeleton  in 
the  family  closet  be  dra.-ged  forth  and  its  ghost  laid." 


1.  Address:   State  Teachers'  Association.  1010, 


■ 


-18- 


Whlle  the  rank  and  file  of  the  public  scho  Is  are  dom- 
inated by  traditional  aims,  the  outlook  is  hopeful.   The  .move 
is  away  from  these  principles.  Under  strong,  progressiva  leaders, 
in  school  communities  favorable  to  changes  Recording  to  social 
needs,  courses  are  being  revised. 

The  practical  aim  is  the  one  accepted,  and  built  upon 
by  the  advocates  of  the  newer  arithmetic.   \ccording  to  this  stand- 
ard arithmetic  is  a  tool  to  be  used  in  the  affairs  of  everyday 
life  v/hen  and  where  numbering  is  necessary.   Its  content  most  be 
drama  from  industrial  and  social  life  and  only  in  no  far  as  it 
functions  in  industrial  practices  and  fulfils  social  needs  has  it 
purpose  and  place  in  the  elementary  school  program. 

The  aim  in  teaching  aritlimetic  is,  first,  to  furnish 
Upll  the  opportunity  to  further  develop  his  ability  in  num- 
bering, second,  to  aid  him  in  acquiring  such  sl:ill  and  accuracy  in 

2 
the  application  of  numbering  as  society  demands. 

1.  Eight  Southern  California  cities:  Santa    ,    ona,  -king 
Beach,  Santa  Monica,  Pasadena,  Riverside,  Redlands,  San  ]  er- 
nardino,  and  the  Teacher  Training  Department  of  the  South- 
ern Branch  of  the  University  of  California  (formerly  the 
Los  Angeles  State  Normal  School)  liavo  band;        Ives 
together  in  committee,  and  are  building  a  com  .on  course  of 
study  founded  upon  practical  aims,  and  developed  along  the 
line  of  social  requirements* 

2.  This,  in  effect,  is  the  purpose  upon  which  the  course  of  study  of 
the  nine  Southern  California  school-units  is  built. 


■19- 


Since  the  aim  is  practical,  the  jlac  held  by  aritlmetic 
in  the  curriculum  can  be  justified  only  by  the  elimination  of  all 
those  parts  vrhich  are  not  useful  to  society  as  a  ./hole.   Turthcr- 
more,  since  nothing  except  what  is  usable  can  enter  into  the  de- 
velopment of  the  child's  concepts  or  notions,  those  parts  of  arith- 
metic v;hich  are  usable  by  the  child  or  can  be  made  to  be  of  use 
to  him  in  his  interpretations  of  his  surroundings ,  must  be  empha- 
sized. 

1.   Note  2  of  page  18. 


-20- 


CHAPTER  III 
I  I  IATIOHS  IN  ARITHMETIC  IIICH  IIAVE  BSEH  SUGCtESTED  OR  MADE. 

The  stirring  need  for  change  in  the  methods  of  instruc- 
tion and  in  the  subject  natter  of  arithmetic  has  boon  felt  in  the 
school  world  only  in  the  past  twenty-five  or  thirty  years.  Un- 
doubtedly the  most  important  factor  in  bringing  about  this  move- 
ment toward  radical  change  is  the  adverse  criticism  of  the  school 
product  by  the  business  world.  As  in  all  real  educational  advance- 
ments the  dynamic  force  came  from  the  users,  the  ones  who  require 
a   Tactical  functioning  of  theories:  the  people.   Thing!  do  not 
grow  well  when  headed  down;  the  moving  force,  the  germination  must 
come  from  society;  the  schools  must  react  to  this  force. 

The  practical,  as  op  ioscd  to  the  disciplinary  virtue  in 
arithmetic  lias  now  passed  the  dividing  of  the  ways  and  is  trying 
to  adjust  itself  to  the  newer  road.   Perhaps  no  subject  in  the 
'nerican  school  system  lias  advanced  throu,:.  nore  winding  ways  of 
experimentation  and  question  in  the  last  two  decades  than  has  the 
subject  of  arithmetic.   Investigators  with  much  enthusiasm,  and, 
we  believe,  much  wisdom,  have  been  making  trial  of  this  and  that 
new  Idea,  weighing  facts,  and  establishing  truths. 

Felix  qui  potuit  rerum  cognoscere  causas, 
we  acclaim  with  Virgil,  if  that  knowledge  comer,  to  bo  his  by  the 
pathway  of  honest  doubt.   Difference  of  opinion  engenders  doubt, 


doubt  stimulates  Investigation,  and  investigation  leads  to  truth. 
Old  forms  of  thought  are  passing,  old  habits  of  nind  give  way  to 
the  new  points  of  view,  and  the  tine  honored  standards  of  instruc- 
tion, vhose  weakness  are  closely  akin  to  wickedness,  arc  being  en- 
tirely lis  lacod  and  the  practical  established  stoutly  in  the  front 
line  of  progress. 

In  reviewing  the  literature  upon  the  subject  of  elinina- 
tlon  one  finds  that  ..hich  offers  nuch  promise,  is  pro- 

verbially conservative,  and  systens  are  so  fixed  by  devotees  that 
they  tend  to  keep  our  schools  behind  the  tines,  hut  the  last  years 
of  the  nineteenth  and  the  first  decades  of  the  twentieth  cer.tury 
are  hopeful  in  outlook.  By  sprinting  we  nay  overtake  the  world. 

Arithmetic,  in  its  primary  conception,  was  wholly  re- 
tleal  in  ideal*   The  Greeks  urged  that  the  common  people  be  in- 
structed in  the  elements  of  logistics,  the  mechanic:.!  phase, 
Plato  mentions  its  usefulness  in  trade.  Arithmetica,  the  theoret- 
ic: 1  aspect,  was  reserved  for  the  philosopher  and  the  scholar. 
Arithmetic  developed  its  pr  ctic  1  connection  because  society 
needed  and  used  it,  and  rot  I  its  raison  d'etre  down  to 

the  middle  age3,  when  it  wa  divorce  :  froa  Its  utilitarian  value 
and  became  a  subject  of  pure  specualtion  for  philosophe:  s  and 
monks.   The  disci.  1 [  ominatc  I  for  several  centuries. 

The  subject  became  replete  with  catch  phr       :  curious  problems 
ingenuously  constructed  by  monks  as  material  for  disputation.   One 
.  m  the  texts  of  to-day  to  find  evidence^'  of  the  ocr- 


sistence  of  this  mediaeval  tendency.   But  through  it  oil,  the  world 

Bd  number  and  used  it.   The  con.orcid  value  of  the  subject  lived 
because  life  required  it.   School  and  practice  were  v/idely  severe  . 
The  one  aristocratically  polished  minds,  the  other  democratically 

doggedly  served  society.   That  the  vocational  phase  of  our 
subject  oust  be  re-established  is  clear  in  view  of  its  history. 

forces,  many  men  and  many  views  of  education  have  combined  to 
make  it  what  it  is  to-day. 

On  examining  an  old  treatise  upon  arithmetic  publls! 
in  Bngland  in  the  sixteenth  century  a  collector  of  old  books  tells 
us  of  the  practical  nature  of  this  old  Etagllsh  treasure. 

The  author,  indulging  In  a  lament  at  the  ignorance  of 
the  people  in  general  which  was  "pitiful  to  talk  of  and  i     .mis- 
erable to  feole,"  of  fern  his  work  ./hich  he  claims  to  be  excellent 
to  aid  all  clarser, ,  especially  the  Mechanics  and  soldiery*   The 
study  is  in  the  form  of  a  dialoeue  made  up  of  "gallant  speeches 
full  of  courtesy."  The  author  begs  the  authorities  "to  be  acquaint- 
ed to  aid  in  paying  of  soldiers'  wages,  a,  ow- 
der,  shot,  muni   ..  ,   nd  instn.             -          lis  into 
'is  problems  are  ty  ac  1; 

"If  a  captain  over  a  band  of  men  did  set  300  plamsres  to 
worke  which  on  eight  houres  did  c  st  a  trcnc:.  of  200  r  les,  I  de- 

Lngfleld|  Lewis:   A  Sixteenth  Century  Arithmetic.   IJLvinr 
Ace,  1G7:315. 


•23- 


mand,  how  many  labourers  will  e  able  with  a  like  trench  in  three 
hourcs  to  entrench  a  canpe  of  3400  rodes?"   Our  present  day  pro- 
portion, the  old  "rule  of  three"  which  was  the  fetish  of  our  fa- 
thers! But  for  the  spelling  and  phrasing  one  might  well  suppose 
the  problem  to  be  taken  fro:;  our  own  state  text  of  1919. 

R.  E.,  Schoolmaster,  is  the  -uthor  of  another  old  book 
prefaced  1G55,  England.   It  is  An  I  den  of  Arithmetjck,  and  con- 
tains  twenty  principles  or  rules.  R.B.  defines  arithmetic  as  "The 
Urt  of  ITumb  ring"  (the  identical  wording  of  Dr.  Ernest  Carrol  I'oore, 
1919)  and  signs  himself  "Of  the  Free  Schools  of  Thurlow."   The  book 
was  compiled  for  the  "furtherance  In  learning  of  Sir  Willi 
Soames*  Hope full  Branch,  William  Soamec,"  Besides  the  twenty  rules 
it  Is  full  of  Latinised  English,  and  is  uilt  upon  the  princi 
of  "Ratio  or  Inequality."  Among  the  rules  are: 

The  Golden  Rule,  (proportion) 

The  Rule  of  Fellowship  (partnership) 

Alligation 

Arithmetical  Progression. 

T>^e   writer  is  struck  by  the  resemblance  between  this  lit- 
tle book  written  throe  and  a  half  centuries  past  and  her  own  child- 
hood text  shell      .   twenty-odd  years  ago.  The  same  rules,  the 
same  exprcs.  ons  abound.  is1  "Ho  e full  Branch"  was 

on  the  threshold  of  learning  by  this  pedantry ,  the  writer  as  a 
"Ilo.'Cfull  "  r  nch"  net  and  struggled  with  the  same  rules,  . 

ernes j  Barli  An  old  Arithmetic.  Acad  .y,  •*. .  >, 


-24« 


hours  and  energy  upon  those  things,  dry,  dull,  deadly  and  useless 

.irucl        ich  bind  us  by  tradition  to  the  past. 

Before  1700,  we  find  little  record  of  instruction  in 
arithmetic  In  the  United  States,  and  such  as  there  was  was  built 
upon  English  customs,  weights,  and  measures  as  the  early  He.. 
Land  settlers  clung  to  the  training  of  the  Old  '.orld. 

In  1G49  Hampton,  Hew  Hampshire,  employed  a  schoolmaster 
"To  teach  to  read,  to  write,  and  to  east  accounts  if  it  be  deslr 

In  1C53  Dedham,  Massachusetts,  had  a  schoolmaster  who 
3  teach  re  I      ,        i  I         nd  "the  knowledge  and  art  of 
arithmetic  and  the  rules  and  practices  thereof." 

In  1789  arithmetic  was  require.:  by  law  In  Massachusetts 
and  :        :hire.   There  had  been  a  few  English  texts  sparingly 

before  the  Revolution,  but  the  manuscript,  the  "cyphering  book" 
and  rule  book  became  quite  common  for  Instruction  of  boys  (girls 

exempted  from  deeper  learning)  after  the  act  of  1789.   The  me- 
chanical, rule-of-thumb  methods,  however,  gave  then  little  advant- 
ovcr  the  uninstructeJ  girls. 

The  author  of  an  old  textbook  published  in  1795,  makes 
an  impassioned  and  patriotic  plea  for  tno  U3e  of  United  States  mon- 
ey, and  the  elimination  of  &oglish  money  and  measures  in  the  - 
Republic.   lie  says,  "Let  us,  I  beg  you,  Fellow-citizens,  no  longer 

1.   Cault,  B«F.i   An  old  text.  BdttC  tlon  20:279. 


-25- 


meanly  follow  the  British  intricate  mode  of  reckoning.   Let  then 
have  their  v/ay  and  us,  ours .  Their  mode  is  suited  to  the  nonius 
of  their  government,  for  It  seems  the  policy  of  tyrants  to  keep 
their  accounts  in  as  intricate  and  perplexing  a  method  as  possibl 
the  smaller  number  of  their  subjects,  then,  may  be  able  to  estimat 
their  enormous  impositions  and  exactions.  But  Republican  money 
ought  to  be  simple  and  adapted  to  the  meanest  capacity." 

We  commend  the  author  as  a  f orward-1 oozing  educator,  for 
he  further  recommends  that  "Insurance,  duties,  commissions,  indeed, 
anything  reckoned  v;ith  per  cents  should  bo  calculated  in  one  head 
and  rule."  We,  in  1921,  are  making  the  same  recomnendation. 

Some  recognition  of  arithmetic  as  a  lov/er  school  branch 
was  made  by  the  early  colleges.   In  1745  Yale  required  arithmetic 
for  entrance;  in  1760  Princeton  required  candidates  "to  understand 
the  principal  rules  of  vulgar  arithmetic;"  and  in  1C07  Harvard's 
admission  statements  required  that  the  entrant  "be  vrell  instructed 
in  the  following  rules  of  arithmetic;"  namely,  notation,  simple  and 
compound  addition,  subtraction,  multiplication  and  division,  to- 
gether with  reduction  and  the  single  Rule  of  Three." 

o 

Among  the  early  New  En  land  texts   which  have  loft  an  im- 
print upon  our  modern  books  for  ^ood  and  ill  is,  at  least,  one  book 
of  English  authorship.  The  Schoolmaster's  Assistant,  1743,  by 
Thomas  Dil«orth,  was  used  extensively  in  this  country  and  held  its 

1.  Erovm,  E.E:   The  Making  of  our  Middle  Schools.  Page  249. 

2.  Sources:  Thomas  Dilworth,  The  Schoolmaster's  Assistant.  Pri- 
vate library  of  U.C  .'..'hoot,  Los  Angeles,  California,  and  I).  S. 
Bulletin  no. 10.  1917 


-26- 


popularity  even  after  the  advent  of  the  comprehensive  work  of 

Nicholas  Pike. 

Dilworth  claims  to  be  practical.  He  has  nothing  to  say- 
regarding  the  nature  and  theory  of  number.   His  notation  and  num- 
eration he  limits  to  nine  digits.   Pike  includes  seventy-eight 
di:its,  duodecillions,  dividing  his  periods,  after  the  English 
fashion,  into  six  digits  each. 

Among  the  topics  included  in  Dilv/orth's  Master's  Assist- 
ant which  educators  are  still  endeavoring  to  eliminate  are: 

I.  Whole  Numbers. 

1.  The  Single  Rule  of  Three  direct,  (proportion) 
- —  Inverse 

2.  Compound  interest. 

3.  Simple  Fellowship,  (partnership) 

4.  Compound  Fellowship,  (partnership) 

5.  Trade  Discount. 

6.  Bxchs 

7.  The  double  Rule  of  Three  (compound  proportion) 

8.  Alligation. 

9.  Progression. 

II.  Vulgar  Fractions. 

1.  Compound  and  complex  fractions. 

2.  Reduction  of  fractions,-  Dilv/oi'th  gives  this  under 
twelve  different  cases,  eaeh  elaborated,  while  addi- 
tion, subtraction,  multiplicat' on  and  divieion  of 
fracti  n3  are  accorded  tnree  scant  pages  with  no  il- 


lustrative  examples  or  explanations  as  to  the  meaning 
or  uf3c  of  fractions* 

III.  Decimal  Tractions.  (This  section  Includes  much  subject- 

ir  not  included  under  that  head  to-day;  the  nc 
decimals  was  not  keenly  felt  until  Unite,:  States  money 
sneral  use* ) 

1.  Square  root. 

.   Cube  root. 

3.  Itole  for  extracting  roots       ere. 

•I.  Reduction:  as,  Reduce  76  yd.  to  a  decimal  of  a  mile. 

•  5.  Applications  of  Percentage  as  separated  to  ics:  bar- 
ter, los:         ,       ,  discounts,  interest  for 
years,  mont1       ,  purchasing  freehold  or  real 
estate,  annuities, 

Among  the  forty  topics  treated  under  decimals  some  score 

have  already  lost  place  in  modern  books;  nine  hold  ,;ood;  and  the 

above  greu        [uestion,  or  has,  in  part,  been  eliminated. 

IV.  Denominate  Numbers. 

1.      Furlong   in  linear  raea.    . 

tod  In  square  measure. 

Troy  weight,    (Dllworth'fl  table   is   similar  to  the 
one  used   to- I 

4.  Clrcu       urc  —  similar  t  to-day. 

5.  Fore  ion  I     -    Lisa  "oney,  In  u:3e  to—.: 
.  apothecaries  weight,  in  use  to-day. 

7.  League. 

8.  Hand. 

0.   Gallons  in  a  barrel,  in  use  te- 
lle very  many  of  4.     L  '  ..  os  given  by 


de- 


arth are  eliminated  fr  nt  day  texts,  the  .ritor 

finds,  by  comparing,  practically  all  in  the  text        .enty-five 
years  aco,  and  further,  finds  nearly  all  in  a  text  by  David  Eugene 

,  A  Grammar  School  Arithmetic,  published  in  1904,  though  in- 
tended for  reference,  chiefly,  In  the  latter  book. 

A  number  of  other  English  editions1  were  printed  in  the 
United  States:  Crocker,  Wlngate,  Sough,  and  the  "first  purely 
arithmetical  work  published  in  the  United  States,"  an  edition  of 
Holder's  Arit:motict  Boston,  1719,  by  J.  Franklin. 

A  rit  lime  tick,  Vulcor  and  Decimal;  with  the  Applications 
thereof  to  a  variety  of  Cases  in  Trade  and  Comr.erce  was  the  first 
American  book  by  an  American  author,  Isaac  Greenwood,  1729.  This 
book  found  snail  place  in  the  schools  and  has  left  little  imprint 
upon  our  texts  for  us  to  commend  or  condemn. 

That  1  3ces.-.ary  "  rt  made  Easy,  James  Ilodder;  The  r cho ol- 
easter 's  ^aslstant,  Nathan  Daboll;  The  Scholar's  Aritlimetic,  TNan- 
iel  Ad  ms,  ay  be  mentioned  in  tracing  this  early  development  of 
the  subject  in  America]  but  the  two  outstanding  ficuros,  the  ones 
v;ho  have  exerted  the  greatest  influence  upon  the  dovolo  anent,  form, 
content,  and  Instruction  of  arithmetic  in  the  United  States,  the 
influence  of  the  one  of  questionable  benefit,  the  other  for  cooclf 
arc  Nicholas  -'ike  and  Warren  Collxirn, 

About  1779  Nicholas  Pike  of  Ilev/buryport ,  Ifassachusetts 
published  his  first  arithmetic,  a  nd  in  1708  his  Uew  and  Complete 


■29- 


System  of  Arithmetic.,   the  first  book  generally  used  in  the 
United  States,  appeared.  An  examination  of  this  book  force-  a  re- 
luctant admiration  for  the  genius  which  could  conceive,  order,  and 
body-forth  the  content.   7,'hat  shape  and  aims  and  agencies  of  edu- 
cation combined  to  produce  him?  "hat  inborn  ability  invented 
this  divine  essence  of  mathematics?  For  it  must  have  Bprung,  "i  - 
erva-like,  from  his  fertile  brain;  practice  was  no  forebear  of  it. 

The  first  408  page*  are  devoted  to  rules, —  a  rule  for 
every  page,  and  sometimes,  though  rarely,  a  demonstrated  problem 
to  elucidate  the  rule  —  to  topics,  to  problems,  to  tables,  to 
specific  directions  of  procedure  and  process;  policies  of  insurance 
arc  given  eight  cases,  the  Rule  of  Three  recognizes  seven  complica- 
tions, while  the  Inverse  Rule  of  Three  and  the  Double  Rule  of 
Three  are  further  sub-divide. 1  and  distinguished  from  the  direct 
eases  in  twelve  different  sections.   There  is  no  overflow  or  cor- 
relation of  knowledge  threatened  here.   Everything  Is  safely  par- 
titioned and  tidily  niched.  Following  the  408  pages  devoted  to 
aritlimetic  are  4  pages  of  "plain"  geometry,  11  pages  of  "plain" 

Biometry a  45  pages  of  nensuration  of  specifics  and  solids  in 
which  are  introduced  practically  all  the  rules  of  geometry,  33 
pages  of  introduction  to  algel .       nod  for  the  use  of  academies, 
and  10  page;;  of  conic  soctl 

The  following  table  of  contents,  srr  raged      .   ike, 
himself,  gives  a  notion  of  the  comprehensiveness  of  the  work. 

1.   Pike,  Nicholas:  Hew  and  Complete  System  of  Aritlimctic.  Reprint, 
irivite  librarv  o    .  .   beet.  Los-  Anrrelos.  dnlifor  in. 


■30- 


Table  of  Contents  of  A  Hew  And  Complete   g.  '        of  Aritlmetic, 
Cy  Nicholas  Pike. 


Numeration:  Pace 

Simple  Addition  17 

Subtraction 20 

Multiplication 22 

Division 23 

Supplement  to  Contraction  in  Molt! plication  34 

Tables  in  Compound  Addition  42 

tion 49 

Problo  Proa  the  Preceding  Rules 5G 

Reduction Gl 

Vtolgar  Fractions  — 70 

Decimal  Fractions 85 

1  ial  Tables 99 

Compound  Multiplication  101 

' Division 106 

Rules  for  Reducing  all  the  Coins,  from  Canada  to 

Georgia;  also  English,  Irish,  and  French  Coins 

Bllai  ,  each  to  the  par  of  .11  the  other  ---  111 

^odecimals,  or  Cross  Multiplication  — 123 

Single  Rule  oi  Three  Direct 125 

The  Methods  of  Making  Taxes,  in  ?  Mote 132 

Single  Rule  of  Mhr  :e  Direct  in  Vulgar  Fractions 13G 

To  Find  the  Value  of  Cold  in  the  Currency  of 

England  and  Virginia 138 

Single  Rule  of  Three  direct  in  Decimals 142 

Rule  of  Three  Inverse 144 

Abbreviation  of  Vulgar  Fractions 147 

Double  Rule  of  Three  147 

Conjoined  Proportion  153 

Arbitration  of  Excliango.;.  155 

Single  Fellowship 155 

Double  Fellowship,  or  Fellowship  with  rime 158 

Fellowship  by  Decimals 1C0 

Practice  161 

By  Decimals 189 

Form  of  a  Till  of  Parcels 191 

Tare  and  Trett 192 

Involution 194 

Evolution 195 

Table  of  Tower.  19G 

Extraction  of  the  Square  Hoot 197 


-31- 


Page 

Ap  11  cation  and  Use  of  the  Square  Root 200 

Extraction  of  the  Cube  Root 203 

Application  and  Use  of  the  Cube  Root  209 

Extraction  of  the  Biquadrate  Root 210 

Of  the  Sursolid,  by  Approximation 211 

Of  the  Rootq6f  all  Power- 212 

Proportion  in  General  210 

Arithmetical  -roportion  217 

Progression 219 

Geometrical  -roportion  234 

_- Progression 23G 

Simple  Interest  251 

By  Decimals  255 

Annuities,  or  Pensions  in  Arrears,  at  lapis  Interest-  204 

Present  Worth  of  Annuities  at  Simple  Interest  205 

Discount 2GC 

.-_  By  Decimals 209 

Tarter 270 

Loss  and  Gain 273 

Equations  of  Payments 281 

P.y  Declmala 203 

ion  or  Factorage  204 

Brokerage 205 

Policies  of  Insurance 200 

Compound  Interest 292 

. By  -.eeinals 294 

. iscount  at  Compound  Interest  299 

Annuities,  or  Penal  ens  In  Arrears,  at  Compound 

interest 300 

Pre  ent  '.orth  of  Annuities  at  Compound  Interest 305 

Annuities  in  Reversion  at  Compound  Interest  309 

Purchasing  Annuities  forever,  or  Freehold  Estates  314 

Table  shewing  the  amount  of  £1  from  1  year  to  50  C10 

Present  worth  of  KL  from  1  year  to  40 —  319 

Amount  of  fcl  Annuity,  etc.  —  320 

Present  worth  of  fcl  Annuity,    etc. 321 

Annuity,  which  El,  will  purchase,   etc.    322 

Circulating    decimals 32;- 

Alllgation  Iledial  — - •-'•20 

Single   Position 334 

Double   Position 330 

Permutations  and  Combinations   339 

A  short  method  of  re  'uci  ir  fraction  to  a 

deci.  al 345 

Of  finding  the  duplicate,  triplicate,  etc.,  Ratio 

of  two  numbers,  wBaose  difference  is  small —  345 

To  estimate  t         as  of  Objects 347 

7ho  "ci  ht  of  Objects 340 


Page 

Hi seel lane ous  Questions 348 

Of  Gravity 357 

Of  the  Fall  of  Bodies 359 

— Of  Pendulums 3G2 

Of  the  Lover,  or  Steelyard 3G4 

Of  the  "./hell  and  Axle  3G5 

Of  the  Screw 365 

-Of  the  Specific  Gravities  of        3G8 

Ta  les  of  Specific  Gravities 3G9 

Use  of  the  r.arometer  in  Measuring  Heights 375 

Table  of  J        ita  of  ;:oney 375 

Table  of  Exchange 37G 

Ditto 377 

Table  of  the  value  of  sundry  ;ioces  in  the  sever:! 

States 378 

Of  Commission 379 

Of  the  net  prcceces,  after  the  commission  at 

2-§  and  5  per  cent  arc  deducted 380 

Table        the  number  of  Days  fr        y  in  one 

th  to  any  day  in  any  other  month 381 

Table  of  the  measure  of  Length  of  the  princi  al 

!  compared  »ith  the  American  yard  -  382 
Tabic  direct!.:  how  to  buy  and  sell  by  the  BBndred 

Weight 383 

Co.  arison  of  the  American  Foot  with  the  Teet  of 

other  Countries  384 

Table  to  c        '-cos  or  Expense*  for  a  Year,  at  so 

much  per  day,  week,  or  month 385 

to  find  Wages,  or  Expenses  for  a  month,  or  day, 

at  so  nach  per  year — -  385 

I      L,  at  6  per  cent  from  1  shil- 

Lin^  to  KL,000,  and  fro:..  1  day  to  a  year,  the  i.  - 

termediate  months  co        of  30  days  c  ich —  38G 

A  Perpetual  Almanack 390 

Tables  reducing  Troy  weight  to  Avoirdupois,  and  the 

Contrary  —  ---  301 

An  account  of  the  Gregorian  Calendar,  or  He.   tyle  —  392 

Chronological  Preble  :s 392 

-e  find  in  which  Century  the  last  year 

is  to  be  Lean  year,  and  the  contr  ry 392 

-  find,  with  r  ether  ye   , 

whether  any  year  be         r,  or  not 393 

;.or.  3.  To  f       Domi  leal  Letter  accord! 

to  the  Julian  method 

according  '.o  the  Gregor- 
ian Method 394 

Problem  5.   '^o  find  the  *Vime  or  Golden      i 395 

Problem  6.   To  find  the  Julian  Epact : 395 

ore'  lem  7.   To  find  the  Or         .act 39G 


- 


Page 

To   find  the   same,    forever 397 

Problem  8.     To  calculate   the  Moon's  agejbn  any  given  day  -  397 
^ro'  lem  9,      To   find  the   tines   of   the   naif  and  full  moon 

and  first  and   last  quarters 398 

Problem  10.      Having  the   time   of  the   Moon's   Bouthi 

given,    to  fi  ge 399 

Problem  11.      To   find   th  f   tha   Moon's   southing  399 

Toblcra  12.      To  find   on  what  day   of  the   week  any  ^iven 

day  in  any  month  ./ill  fall 400 

Problem  13.      To   find  the   Cycle   of  the   Sun 401 

Table   of  the  Deminical  Letters  accord 

to  the    Cycle   of  Sun 402 

Problem  14.      To   find   the  Year  of  the  Dynonisian   Period  —  402 

Problem  15.     To  find  the  year  of  Indiction 402 

m  16.      To  find  the   Julian   "eriod 403 

"roblcr.  17.      r,o  find  the  Cycle   of  the   fun, Golden  Number, 

and  Indiction  for  any  current  year 403 

Problem  18.     Having   the    Cycle   of  tic    Sun,  Golden  'lumber, 

and   Indiction,    to  find   the  year  of  Christian  Era  —  404 

Problem  :.C.  ater 404 

Problem  20.      To   find  on  what  day  Easter  will  hap  ten 404 

er  fro:;  the  Year  1753  to  4199 40C 

The  Use  of  logarithm! 407 

This  table  i3  inadequate  to  give  the  full  significance 
of  the.  content a  however.  To  illustrate:   Under  the  Mead  "Rules 
for  reducing  Federal  coin  and  currencies  of  the  sever  1  United 
States,  al3o  English,  Irish,  Canada,  Nova  Scotia,  Livroc,  Tournois, 
and  Spanish  milled  Dollars  each  to  the  par  of  the  other"  Mr.  Pike 
gives  7C  rules  for  the  exchange  of  coin  among  the  Mtates  alone,  and 
then  treats  the  foreign  nations  with  equal  consideration. 

As  a  general  reference  book,  or,  by  selection,  a  text  for 
special  technical  schools  the  1  Ly  valuable.   Son0 

of  the  rules  are  quite  unintelligible  to  modern  students .   Under 
Trott  and  Tare,   ike  says:  "Deduct  the  tare  and  trott,  divide  the 
suttle  by  1G8  and  the  quotient  ..ill  be  the  cloff ,  which  subtract 


■:-;• 


from  the  Buttle  and  the  remainder  will  be  the  neat."  Evidently, 
"trett,"  "tare,"  "suttle," "cloff"  and  "neat"  were  of  the  vocabu- 
lary of  th  day,  and  thia  rule  for  the  wei  hing  and  handling  of 

merchandise  was  useful  to  a  group  of  traders.   One  can  hardly  con- 
ceive of  these  rule  bell  g  useful  to  the  average  citizen  in  his 
daily  life,  and  even  the  advanced  students,  for  whom  the  text  was 
intended,  could  have  found  little  with  which  their  experience  cor- 
responded. Yet,  shall  we  criticize  ^r.  Pike's  work  while  our  own 
texts  retain  surveyor's  measure  and  apothecaries  weight? 

Penjamin  best,  in  a  criticism  of  the  book  says,  "The 
volume  contains,  besides  what  is  useful  and  necessary  in  the  com- 
mon affairs  of  life,  a  great  fund  of  amusement  and  entertainment. 
The  mechanic  will  find  In  it  rmich  that  he  nay  have  occasion  for; 
the  lawyer,  me  reliant  and  mathematician  will  find  an  ample  field 
for  exercising  their  genius." 

This  book  was  the  gem  puzzle  of  its  day.   II  ny  .leasing 
and  diverting  questions  are  included  in  its  lists  of  preblei  . 
These  problems  are  connected  with  questions  as  to  the  number  of 
changes  which  can  be  rung  on  a  chime,  how  many  different  positions 
a  person  can  assume  at  a  dinner  party,  .        v  riations  can  be 
made  of  the  alphabet,  and  many  others  which  reflect,  the  influence 
of  mediaeval  dis  utation.   c  find  the  forbear  of  our  puzzle  of 
the  fox,  the  gect;e  and  the  bags  of  corn  and  the  complications 
arising  from  carrying  then  across  the  river  two  by  two,  which 
touches  a  rcsoonsivc  association  in  the  minds  of  the  students  of 


■35- 


the  early  nineties.  Then,  in  geometrical  progression,  is  that  ap- 
parently foolish  bargain  of  tue  merchant  vho  sold  39  y >rds  of  fine 
velvet  for  2  pins  for  the  first  yard,  6  pins  for  the  second  yard, 
18  pins  for  the  third,  etc.  which  turns  out  most  profitable  to  the 
seller  after  all.   One  of  our  fairly  modern  books  ha3  a  problem 
wherein  a  crafty  person  offers  a  hundred  acres  of  land  at  tvo  pen- 
nies for  the  first  acre,  6  for  the  second,  etc.  which  the  writer 
has  used  recently  as  a  diversion  to  the  delight  and  v onderment  of 
her  pupils . 

Pike's  text  ran  through  five  edit:^ns,  the  fifth  printed 
in  1832,  and  despite  the  reaction  created  by  Warren  Colburn's 
First  Le  s  c ons ,  is  the  book  which  is  most  responsible  for  the  ne- 
cessity for  extensive  eliminations  to-day. 

The  great  reformer  in  American  arithmetic  was  Warren 
Colbum.   He  made  a  protect,  and  a  vigorous  one,  against  the  deep- 
root  d  evils  of  the  mediaeval  and  English  notion  of  teaching  num- 
ber as  symbols,  and  of  clinging  to  definisions  and  ruler,.  He  en- 
deavored to  seek  out  and  seize  upon  the  instincts  of  the  child  and 
use  these  as  a  factor  in  Lis  educatin.   On  reading  his  masterly 
address  delivered  before  the  American  Institute  of  Instruction  in 
Boston  in  1830  on  Teaching  of  Arithmetic,  one  realizes  he  possessed 

1.  Reprinted  in  Elementary  School  Teacher,  v.  12,  (1911-12) 
pages  463-480. 


•36- 


an  Insight  into  the  educative  process  far  ahead  of  hlfl  time,  and 
which  wo  are  but  now  coining  to  realize. 

Colburn  was  avowedly  a  follower  of  Pestalozsi.   In  his 
preface  to  the  second  edition  of  his  First  Lessons  is  a  lengthy 
tribute  to  Pestaloazl  which  gives  clear  evidence  of  his  acquaint- 
ance of  the  Postaloznian  plan,  although  he  was  Largely  independent 
in  his  conceptions  and  in  the  carrying  out  of  his  theories.   Mr. 
Thomas  Sherwin,  princinal  of  the  hi        I,  Bee  ten,  (1030)  says: 
"I  regard  Mr.  Colburn  as  the  great  benefactor  of  his  age  with  re- 
spect to  tho  proper  development  of  the  mathematical  powers.   Pesta- 
lozsi, indeed,  first  conceive.':  the  plan;  but  Mr.  Colburn  realized 
the  plan j  popularized  it,  and  rendered  it  capable  of  being  ap  lied 
to  the  humblest  mediocre oy.   Indeed,  I  regard  First  Lessons  as  the 
ne  plus  ultra  of  primary  arithmetic." 

First  Lessons  in  Intellectual  Arithmetic,'  published 
first  in  1821  achieved  almost  incr  text.   *t 

was  followed  by  The  f.equel  to  First  Lessons  a  year  later.   Fiivt 
.  .  - ■ ! s  is  rerlly  a  "L'lnimum  Essentials*  of  arlthtaetie.  Mr,   Ccl- 
burn's  idea  wa3  to  eliminate  ail  useless  material,  to  teach  by  the 
use  of  concrete  objects,  and  to  connect  each  question  vith  tho  ac- 
tivity of  the  child  himself. 

L.   Elementary  School  Teacher,  vol.  12,  pago  424. 

2.  University  of  California  Library  -  renrint. 

3.  University  of  California  Library  (original,  edition  of  1034) 


-37* 


A  comparison  of  the  table  of  contents  of  First  Ler 
with  that  of  Nicholas  Pike's  Nq\y  and  Complete  System  of  Ari  time  tic 
is  illuminatj 

Table  of  Contents  of  Warren  Colburn1 s  First  Lessons. 
Part  I. 

Sec.  I.    Addition  and  subtraction. 

Sec.  II.   Multiplication. 

Sec.  III.   Division.  -  Idea  of  fraction  introduced. 

Sec.  IV.   Fractions;  multiplication  of  an  integer  by  a  fraction. 

Sec.  V.    Principle  of  fractions  applied  to  larger  nu   i  . 

Sec.  VI.   Division  of  an  Integer  by  a  fraction. 

Sec.  VII.   Compilations  of  preceding  and  multiplication  tabic  from 
10  x  10  mp   to  10  x  20. 

Sec.  VIII.  Reduction  of  fractions  to  higher  ter      Integers 
to  fractions. 

Sec.  IX.   Uultiplical       a  fraction  by        9r« 
.  X.    Mostly  dril^on  Sections  IV  and  IX. 

Sec.  XI.   Division  of  one  frao       another. 

Sec.  XII.  Fractions  written  in  fractional  form. 

Sec.  XIII.  Reduction  of  fractions  to  a  comma       L  lator;  addi- 
tion and  subtraction  of  fractions. 

Sec.  XIV.  Division  of  fractions  by  integer:         I  Lieation 
of  a  fraction  by  a  fraction. 

Sec.  XV.   Mvisj  »n  of  integer.:  by  fractions  aa         I   by 
a  fraction. 

Colburn  specifically     3  ns  the  cli        of  The 
Rule  of  Three  (proportion)  of  Cube  Root,  and     oaxe 

:ts  that  Denominate  Numbers,  Perce:     ,  [nterest  and  He 
ration  should  not  bo  taken  as  bases  for  separate  chapters  or  0VGn 
distinct  to  vie.;. 

1.   Colburn  d  Me  of  contents  in  the  First  Los- 

sons,  and  U        sen  made  from  a  study  of  the  material 

en  in  the  various  sections. 


-r,8- 


It  is  cl-  Interesting  fact  to  note  that  in  1821  '.7arren 
Colburn  re conme nded  tho  elimination  of  Square  Root  from  the  c il 
tary  school  text  and  that  the  writer ,  after  a  lapse  of  one  hundred 
years,  should  find  it  necessary  to  make  the  sane  recommendation. 

Follov/ing  Colburn  came  the  books  of  Jose  h  Ray,  books 
built  along  the  line  of  Pile  rather  than  Colburn,  and  from  the 
waning  of  Colburn' z   influence  to  the  latter  part  of  the  nineteenth 
century,  arithmetic  war  taught  as  a  mental  discipline.   The  body  of 
natter  accumulated,  tr  '  lng  what  lad  o^e  before,  and 

a  desire  for  new  disciplinary  material  adding  to  the  ball:. 

Throughout  this  static  period  people  aeeepted  rather 
than  investig  ted.         served  for  information  and  that  o 

.  1':    jen  -h,  nd.  It  ia  relate"  of  a  learned  judge 
.e  once  praise       ..ring  witness  in  these      ,  "You  are 
entitled  to  great  s        ir.   You  BUS1  ito  alns 

with  yourself*   "do  man  could  naturally  .    Our  great" 

'  arithmetic  COW  -Tort. 

While  there  e*aa  no  concerted  action  te        lifying 
and  relating  numbering  to  use  -hiring  these  .        i  pa,  one  finds 
protests  against  the  conventional  teaching  a  o  the 

dogma  of  formal  discipline,   A  number  of  eri1         Li  id  pleas 
that  schools  avoid  more  empirical  rules  arid  keep  in  view  tho 
practical  and  commercial  value  of  arithmetic        t  of  computa- 
tion, rate  according  to 
trade-rules  which  they  do  not  understand  arc  quite  at  loss  to  help 


■  ■ 


themselves  i:  .      :t,  although,  by  this  method 

(disciplinary)  the  scholar  nay  be  ."oil  prepared  for  .any  computa- 
tions which  he  wlH  have  occasion  Tor  in  practical  life,  yet  lie 
will  be  quite  at  a  loss  hew  to  help  himself  whenever  a  case  shall 
cone  up  to  rhich  he  cannot  a  ply  his  rule  exactly  as  he  has  1 
to  use  it."2 

Little  farther  was  done  In  the  way  of  simplifying  arith- 
metic until  1387  when  President  Fr  i     ".   lker  attacked  the 
problem  in  the  Boston  schools*  and  his  investigations  resulted  in 
a  recommendation  by  the  school  committee  that  the  fol       ubjeeti 
be  eli        from  the  course  of  study i 

1.  Mensuration  of  unusual  surfaces  and  sc 

2.  Comooiind  proportion. 

_  id  Interes    . 
-1.      Equation  of   payments* 
5.       ■ 

G.     Metric  System*  , 

7.      Comp   a 

th     twentieth  century,   signifi- 
cant rithm  advocated  by  certain  note:  educat- 
ors  throughout                X  and  the  Kiddle     est.      I  .  lr  pur- 
pose to  search  ou1             racticatl  In  sritheetie  and  tc  o:nit  all 
other  material.     In  1003  c.  "...   Stone   sent  out  s  (zuestioonalre 

1.  Peacock,  (-;-. :     Educational  Value  of  Arithmetic.     London 
Quarterly,  1058. 

2.  Bernard's  Journal  of  Education,  vol.0   (18G0) 

e   170. 

3.  Ifossup,       .Iter  A.:      Educational  Research,   School  and   Society. 
July  24,   1915.      Page  137. 


•40- 


to  business  men  of  Indianapolis,  requesting   their  opinions  on  the 

utilitarian  value   of  the   work   ~iven   in   arithmetic.      The   replies   in- 
dicated  that  certai:.  to  ics  had  absolutely  no  use   In  the  business 
world. 

Dr.  lurry    in  an  addr...  Ions 

are  advisable   in  the  present  course   of   study,   and  what  should  be 
the  basis    for  the    same",    delivered  in  1904  befoi  ti  nal 

Department  of  Superintendence,  recommended  that  the  fc  . 
topic  tic: 

1. 

2.  Tre 

3.  Examples  '      Ltude  and  Tine,  except  the  very 

simplest. 

4.  The  furlong  In  li  -  Lsure. 

5.  The  rood   in  square     measure. 

G.     The  dram  quarter  In  avoirdupois  weight* 

7.      The  surveyor's   tal 
3.      Tal  le    on  fol  per. 

9.      All  problems   in  reduction,   asccn!inc     -:!  fcsc ending, 

. 

10.  The   G.  C.  C.  as  a  separate  topic,  but  not  practice 

in  detecting  divisibility  by  2,  3,  5,  and  10. 

11.  All  common  froctior         those  of  a  very  low  de- 

nomi    on 

12.  Ml   work  with  the    •-•.   C.    '".   except  of  such  very  Ion 

i  .  those   Ju 

13.  Cttons  r-  bo    "c. 

14.  Compound  pro 

15.  Perce   t  S    separate   topic,   with  its   cases. 
1G.     True  di     ount, 

17.  "est   problems    in  compound   interest,    and  all    in  annual 

18.  Problems   in  partial      ay  ierit  e   of  a  very 

le  hind. 

19.  The  same  for  comnis~lon  and  broker  ;ej  for  txample, 

all  problems  involving  fractions  of 

20.  Profit  and  [  i  cial  to  ic. 

21.  Equation  of  payments  - 

proved  bankin  faclll  I 


-41- 


22.  Partnership  -  nado  unnecessary,  in  the  old  oc.   , 

by  stock  conpani 

23.  Cube  Root. 

24.  All  algebra,  except  cue'        ace  of  the  equa- 

tion as  la  directly  helpful  in  Arithmetic  and 
in  other  subjects  in  the  shcool  life  of  the 
pupil* 

In  addition  to  all/of  thee,  arithmetic  may  be 

omitted  r.  a  separate  study  throughout  the  firct 
year  of  school  on  the  ground  that  there  is  no 
need  of  it  if  the  number  incidentally  called 
fear  In  other  work  1c  ropcrly  attended  to.1 

In  1904  Joseph  v.  Collins  of  the  ::tate  Normal  School 
_  .o  very  definite  recommends.'- 
[a  i   -  eussion,  "The  ruperintond- 
c:.t  and  th     .  i  of  Study"  be  aay*i  ''If  instruction  in  schools 
in  arithmetiCj  is  to  bs  ^>ro^  ht  u-j   to  a  place  \diere  it  will  be  rc- 

Ld  it  will  have  to  include  only  oper- 
as . " 
He  .  .  pendents  ge  throv.        cxt 

books  they  nay  have  in  ur;^        ,1^  blue  pencil  and  sea  to  it 
Llevln  omittedi 

1.       ...  ic. 

.  "'.   .   .   .  •  nto  be  nit  with  addi- 

ti  n  of  fr 
3.   Longitude  ai  j), 

(put  .«'! 
5.  rrc, 

C.   A  vast  deal  In- 

ate 

He  says,  "All  of         .  , 

'-•  .  .'.en  of  conpound  denominate  numbers  shoul  - 

1.   UcUurrV  >   Prank:  :  rocccuiivjc  of  the  Nati  nal 

Bducati  Lation,  1004  pace.?.  10-1-2  02. 


■    - 


lete,    as  also  all     rob  lens  '  clvo   qjantitic.  '   in 

more   than  tv/o  denomination::;.      Such  a  problem  as:      "Reduce  2ni. 
3rd.    5yd.    8ft,    5   in.    to  inches"   is   aa   absur  "  eto- 

ry.      It   Is  evidently   the  product  of  soae  school;;  ' 

Invent        .        uch,   hawnror,   ic    the   force  of  cuntom  that  numerous 
pro'  lens  of  t  o  be    Pound   in  most  of  the  ice. 

of  this  day.1     The   superintendent  should  dra  inell 

through  all   the   tables   of  u;bcrc   except  .is 

uare,    cubic,   dry,    liquid,   end    '  ires, 

all  problems  under  them."    "'  i   of  typic 

analysis   ^.nd  proportion  bciu£,   "strictl;  ,  raetleal*" 

i'r.    Cel   !  1  I  i  try 

in  the  upper  elements  ry  0rade...      "Let  rjeomet  I   for 

p 
iration,   and    Let   the     \izzlo  pr<  laco  J 

In  1909  G.    Li.       '_  .  a    study  of  the   social 

Of  aril.  ...  -i v.; .  n      published  his   fi.                       the 

Sixtc           V     r  Book  of  l  nal  Society   for   Mid  Study  of  I 

cation,   do   s;  . 

"In  connection  .,i  .1 

in  Arithmetic  •  .,  Cecnersvl  ,  a  fc\7  y      ;,  an  ct- 

tempt         to  get  the  judgment  of  the  business  c  on  a 

hnetlo  to_  ic.  .  ro.-.ult  of  this  cc  .       , 

1.  The  writer  ad!s,  "And  thi:.   ,  ote  the  Cali •"■        to 

Text  of  1919. 

2.  Collins,  Joseph:  The  Superint         rid  the  Course 
Study.  27.  33-09. 


-43- 


business  nen  of  the  city  voted  to  o  .it  the  foil         les 
fron  the  aritlunotic  coui^ses: 

1.  Troy  Weight. 

2.  Apothecaries 

3.  Lonritude  and  Tine. 

4.  The  surveyor's  table. 

5.  The  Greatest  Common  Divisor . 

6.  The  Least  Common  Multiple. 

7.  Complex  Fractions. 

8.  Cube  Root. 

9.  Compound  Fractions. 

10.  Foreign  Exchange. 

11.  Compound  Proportl 

12.  Time  discount. 

13.  Case..  2  and  3  in  Percent 

14.  Compound  Interest. 

15.  Partial  Payments. 

16.  Partnership.       1 

In  1910  the  Baltimore  Scho  1         n  criticised  the 
time  expenditure  and  to  ic  1  emphasis  in  arithmetic  and  suggested 
alterations,  adjustments  and  eliminations  be  made  to  meet  the  needs 

2 

of  the  business  world  and  in  1919  recommended  tic  so  eliminations: 

1.  G.  C.  D.  and  L.  C.  H.  of  lar^e  numbers  etherwl 

than  by  factori   . 

2.  Fractions  with  lar^e  and  unusual  denominators. 
J.   Complex  and  Compound  fr  ctions. 

4.  All  measures  not  actually  In  use  in  the  community 

at  lar^e:  troy,  apothecaries,  dr 

5.  Reduction  of  decimals  to  com  .on  fractions,  and 

decimals  beyond  thousandths  should  receive  little 
basis. 
G.   Circulating  decimals.   The  to:)ic  should  be  studied 
as  a  part  of  infinite  series  in  alcebr-  . 

7.  Square  root  and  cube  root  exc 

8.  Profit  and  loss  as  a  separ       ic. 

9.  True  discount.  "  ank  discount        n  its  place 

entirely. 

Lis  on,  G.::.:   Sixteenth  Yearbook  of  the  Rational  Socioty  for 
the  study  of  Education.  Chapter  VIII,  Page  128. 
2.   Bureau  of  Bducatl  n  Bulletin  1911,  No.  4,  na._.c  70. 


-:  •- 


10.  Partial  payments  in  tho  forn  of  state  rules 

irregular  indorsemen 

11.  Equation  of  payments. 

12.  Compound  proportion  has  been  largely  replace  J  by 

unitary  analysis.  Simple  proporti  in  is  of 
sorae  importance  but  is  beet  treated  as 
equality  of  tv/o  fractions. 

13.  Business  problems  which  do  not  conform  to  the 

usage  of  the  day. 

14.  Large  numbers  and  exorcises  involving  many  nun-  , 

bers  should  also  be  excluded  as  a  rule. 

In  1911  the  American  Committee  :io.  1  of  the  Internation- 
al Committee  on  the  ne  ChJ        thematics  reported  on  Mathe- 
matics in  the  Elementary  Schools.  This  Committee,  through  its  in- 
vestigations, found  that  school  and  business  people  felt  an  urgent 
for  simplifying  courses  of  study: 

1.  By  naing  small  numbers  in  work  in  arithmetic. 

2.  By  eliminating  topics  that  are  unduly  confusin  . 
By  eliminating  obsolete  problems,  topics 

processes* 

In  1912  or  1913  Dr.  Walt  <r  A.  .... 
tan  determined  to  submit  the  problems  of  elimination  nd  en- 
richment of  the  curricula  to  the  su        lents  of  all  cities  of 
tee  with  a  p'            ,          or  over.  This  inves- 
ts reported  in  the  Four  to-. _     the  nation- 
al ociety  for  the  otudy  of       Lon,  indicates  a  decided  ten 
ey  among  ths  superintendents  of  these  cJ  :      :        gen- 
eral proposition  of  either  eli                Less  attention 

1.  Course  of  Study  -  Baltimore  County.  1919.        .1.0--.  13. 

2.  Jessun,  Palter  '. :  School  and  Society,  2. 


-45- 


to  the  topics  originally  sugge         I  .   c;:urry.   They  recom- 
mended that  these  1        to,  or  chanced,  by  further  study.   The 
reports  show,  also,  e  stron  the  superinten  lo^ts  in 

favor  of  more  attention  to  economic  and  bus!        llcati  ma   of 
arithmetic.   From  this  return,  Dr.  Jeesu  I  that  an 

economy  oT       [  _;ht  be  effected  by  the  elimination  of  the  follow- 
ing topics  from  the  elementary  course  of  study: 

1.  Apothecaries  Weight* 

2.  Alligation. 

3.  Aliquot  Parts. 

4.  Annual  Interest. 

5.  Cube  Root. 

6.  Casos  in  Percentage. 

7.  Compound  and  Complex  Fractions  of  more  than  two 

digits. 

8.  Compound  proportion. 

9.  Dram. 

10.  Foreign  Uoney. 

11.  Folding  naper. 

12.  The  Ion-  method  of  G.   .   . 

13.  Lo  .  d  Time. 
14*  Least  Common  Multiple. 

15.  Metric  System. 

16.  Progre  ision. 

17.  Quarter  in  Avoirdupois. 

18.  P.educti   i  n  two  Bt< 

19.  Troy  Wei 

20.  True  Discount.    .. 

21.  Unreal  .  1 

(■"•hart,   p.HG  -  Fourteenth  jok.} 

t  writers  on  the  teaching  of  Arithmetic  suggest 

many   eliminations.      In  Brown  and  Co  '      ___. Arith 

.    follo'./i:  Lnations  are   recommended: 

1.      Jessup,  W,   A. and  Coffman,    L.    D. :    The   Superrl  Arithme- 

tic.  Chapter  1.  -13. 


-46- 


1.  G.  C.  D. 

2.  L.  C.   . 

3.  All  obsolete  tables  in  denominate  numbers  and 

all  tables  that  are  of  use  to  the  specialist 
only . 

4.  Long  or  unnecessary  reductions. 

5.  Circulating  decimals. 

G.      All  appll cations   of  percentage    that  do  not   conform 
to  present-day  practices. 

7.  True  discount. 

8.  Equation  of  Payments. 

9.  Cube  Root. 

10.  Progression. 

11.  Compound  Proportion. 

12.  Problems  which  require  long  and  involved  solutions. 

13.  All  fractions  except  those  used  in  every-day  busi- 

ness life  - 

a.  Long  fractions. 

b.  Complex  fractions. 

14.  Partial  payments. 

15.  All  topics  Which  time  or  changed  social  conditions 

have  rendered  obsolete. ^ 

In  1915  the  Committee  on  Eliminations  of  the  Iowa  State 
Teachers'  Association  rocora.  ended  a  sv/eoping  elimination  of  obso- 
lete and  useless  topics  and  materials  from  the  common  branches.  A 
second  report  of  a  more  definite  nature  was  made  in  191G  in  which 
this  Committee  recommended  that  the  following  cli  ;inations  be 
made  in  ari  time  tic: 

1.  Long  method  of  greatest  com.  ion  divisor. 

2.  Host  of  lowest  com  on  multiple. 

3.  Lorn  ,  confusing  problems  in  common  fractions. 

4.  Long  method  of  division  of  fr  cti  ns.  (Always  in- 

vert and  multiply.) 

5.  Complex  and  compound  fr  cti  .is. 

G.   Apothecaries'  weight,  troy  weight,  the  furlong  in 

lonr  me,. sure,  tloc  rood  in  square  measure,  dram  and 
quarter  in  avoirdupois  weight,  the  surveyor's 
table,  the  table  of  folding  .apcr,  t  Dies  of  for- 
eign money,  all  reduction  of  more  than  two  st 

1.  Brown,  Joseph  C.  and  Coffmrm,  Lotus  D.   How  to  Teach  Arith- 
metic.  Pages  117-118. 


-47- 


7.  Host  of  1  ngitude  and  time. 

3.  Cases  in  percentage.   (Uakc  oi.   |     y  using  x 

I  e  uation. ) 

9.  True  discount. 

10.  Most  of  compound  and  annual  interest . 

11.  Partial  payment,  except  the  simplest* 

12.  "'rofit  and  los  as  a  separate  topic. 

13.  Partnership. 

14.  Cuba  root.  L 


David  Eugene  Smith  in  a  chapter  on  ari timet ic  in 

Te  ■.chir.j  Elementary  Scho  . j  .■■  Jts  by  L.  W«  Rapoer  and  o' 

recommends  that  relative  values  In  arithmetic  be  considered.  He 
would  eliminate: 

1.  Fractions  with  larae  denominators. 

2.  Division  of  a  fraction  by  a  fraction. 

3.  Multiplication  of  a  friction  by  a  fraction. 

4.  Cases  in  percent;' 

5.  Subject  natter  found  of  little  value  in  the  business 

vorldt 

Square  root. 

Cube  root. 

Progress 

Equation  of  laymcnts. 

Proportion.2 

The  Teaching  of  Ari timet ic  published  by  John  Charles 

Stone  in  1918  recommends  the  elimination  of: 

1.  Cre  test  Common  . ivisor. 

2.  Addition  and  subtraction  of  fractions  ,7itli  large 

or  unusual  denominators. 

3.  Least  Common  Multiple. 

The   ore  complex  form.:  of  oomplez  frictions. 
5.   Obsolete  tables  and  those  U3ed  in  aepclallzed  voca- 
tions. 

ilson,  G.   .      >ort  of  the  Committee  on  Elimination  of 
Subject-matter  to  the  Io..        .'eachers  As-.ociat' 
2.  Smith,  David  Eugeno:   Teaching  Elementary  chool  Subjec'  . 
Raoecr  and  Others.       I  207-249. 


-48- 


6.  Impractical  ro  luctions  in  denominate  numbers. 

7.  Addition,  subtracti  on,  multi  >lication,  and  divis- 

ion of  compound  denominate  numbci  . 

8.  Tho  present  type  of  inverse  iroblons  in  fracti  ns 

and  percent; 

9.  The  various  short  methods  of  fin  line  interest. 

10.  An  inverse  problems  of  interest. 

11.  Partial  payment  . 

12.  Annuo.  1  intore  t. 

13.  Undue  emphasis  upon  the  discounting  of  interest- 

be  aring  notes. 

14.  True  discount. 

15.  Partnership. 

16.  Proportion  as  a  gee    -   'A\od   of  so 

17.  Foreign  and  domestic  exchange. 

13.   The  measurement  of  uncommon  areas  and  volumes. 
19.  Square  root  and  the  Pythagorean  Theorem. 
29.  The  metric  system.1 

This  study  seems  to  justify  the  conclusion  that  up  to 
the  twentieth  century  tradition  tended  to  keep  subject  matter 
which  had  once  been  added;  that  net?  subject  matter  was  much  more 

ly  incorporated  than  obsolete  matter  discarded*  This  result- 
ed in  a  mass  of  material  which  sag  cumbersome,  impractical,  much 
of  it  utterly  useless.  But  v/ith  the  beginning  of  tho  tv/enticth 
century,  certain  unprejudiced  searchers  after  truth,  realizing 
the  necessity  for  radical  change,  began  a  vigorous  campaign  for 
the  elimination  of  this  useless  material  nicked  up  in  the  devel- 
opment and  progress  of  the  subject.   There  investigators  claimed 
the  air.  of  arithmetic  is  to  enable  one  to  do  intelligently  the 
common  work  of  the  world,  and  they  proceeded  to  discover  fro.  the 
world  what  portions  of  the  subject  are  serviceable  in  interpreting 

1.  Stone,  John  Charles:   The  .otic.   1918. 

Ilote:   The  r,tone-;'J.llis  texts  are  faitliful  to        i  1. 


•49- 


our  surroundings  from  a  number  point  of  vie.,,  and  to  eliminate 
the  rest, 

y   forces,  many  men,  and  a  coiUMendable  amount  of  scien- 
tific investigation  have  combined  to  make  the  subject  what  it  is 
to-day.   '.That  we  have  achieved  thus  far  has  serve  :  ,  ut  to  in- 
crease our  interest,  and  to  urge  us  to  further  investigations 
of  every  phase  of  arithmetical  work. 


•50- 


CIIAPTER   IV 
MINIMUM  ES   ENTIAIS   IN  ARITHMETIC. 

"The  minimum  essential  in  arithmetic  is  the  ability 
on  the  part  of  the  individual  to  do  practical  calculations  such 
as  are  needed  by  the  average  citizen  In  his  daily  life."1 

Socrates  "cursed  as  impious  him  who  first  separated 
the  just  from  the  useful,"  and  we  add,  "him  who  first  separated 
common  sense  and  ari  time  tic." 

Purine  the  past  two  decades  forward-looking  teachers 
and  school  administrators  have  been  concerning  themselves  with 
the  Tuestion,  "V/hat  is  essential?"  and  have  been  laboring  to  re- 
lieve the  commonly  accepted  curriculum  of  its  unjust  burdens. 
Pal  lie  opinion  itself  is  makinc  a  stronr  ap  teal  that  the  arithme- 
tic taught  La  more  in  tune  with  life.  In  the  expression  of  dissat- 
isfied business  men  there  is  very  little  an        of  let  ils 
in  the  solution  of  the  problem,  but  there  is  a  stern  demand  for 
results.   Schools  have  been  tardy  in  response,  but  leaders  have 
been  studying  to  determine  what  parts  of  arithmetic  have  definite 
utility,  and  are  providing  special  opportunities  fop  giving  ex- 
clusive attention  to  those  aspects  of  them. 

President  Francis  ' .   alker  of  the  Massachusetts  In- 
stitute of  Technol',         the  crusade  in  1807.   Re  declared 

1.   Report  of  Committee  on  Course  of  Study  In  Arithmetic.   Los 
Angeles  St  to  Normal  School,   1919. 


that  "a  f  lsc  arithmetic  has  grown  u?  which  has  largely  crowded 
out  the  place  of  tiiie  arithmetic  — -  The  most  jagged  fractions  such 
as  would  hardly  ever  be  found  in  actual  business  operation,  e.g., 
ll/20,  or  13/27  are  piled  one  on  top  of  another,  to  produce  an  un- 
real and  i       Le  difficulty;  the  child  having  been  furnished 
with  such  an  aritlimctical  monstrosity,  is  set  to  multiplying  or 
dividing  it  bj  another  'compound  and  complex  fraction'  as  unreal 
and  ri  liculous  as  itself*  All  this  sort  of  thing  in  the  teach- 
ing of  young  children  is  cither  useless  or  mischievous.  The 

Inst  the  existing  course  of  study  Is  th  t  it  is 
up  of  exercises  which  arc  not  exerciser,  in  arithme- 
tic at  all,  or  principally,  but  are  exercises  in  logic,  and  second- 
ly, that  as  exercises  in  logic,  these  are  useless  or  mischievous 

General   ,  '   ..ot  universally  speaking,  whatever  in  education 

,     ireng.    I!oro  than  thirty  years  have  gone  by  since 
then,  and  the  arithmetic  t -tight  is  still,  for  the  most  part,  the 
Ltional  thl  .   V.zcr   found  so  meanl      .  We  use 

a  year  or  nor  to  teach  fractions.  Let  bch  him- 

self carefully  for  a  month  and  discover  when  he  had  occasion  to 
add  2/5  and  3/4,  or  subtract  2/5  fron:  3/4,  or  to  multiply  a  fr  c- 
tion  by  a  fraction.   I  challenge  anyone  to  discover  an  occasion 
in  life  for  dividing  a  fraction  by  a  fraction.1^  Yet  in  practic  1- 

tlker,  Francis  ki      Arithmetic  in  the  J  oston  r.cho  Is,  "  cadomy 
(Syracuse)  1087.  Pago  433. 
2.  An  Educ  erintend 

ri  cipals  a  f  e::oericnce  accepted  this  challei 

but  faile '. ,  duri  ,  to  find  le  itimate  oc- 


ly  any  text  book  used  in  our  Ian J,  we  fin  1  this  sort  of  problem: 
o/29  x  ll/l9,    1G  5/23  -4-  15/43. 

Too  often,  even  now,  children  are  worried  over  the  number  of 
cubic  inches  In  a  gallon,  compound  proportion,  cube  root,  aliquot 
parts,  such  problcus  as,  "If  the  principal  is  £u43. 87,  the  time 
5  years  7  months  and  25  days,  and  the  amount  'US729   what  is  the 
rate?"  This  Insane  query  is  from  the  writer's  childhood  text: 
''Ho'.'  many  gills  in  2  hogsheads,  31  gallons,  3  quarts,  and  2  pints:" 
A  liquor  dealer,  retailing  in  gill  quantities,  may  have  found  oc- 
casion for  such  computations,  but  we  coul  :  hardly  cl       1  an 
essential  sir.ee  the  Volstead  Act.   From  the  addition,  subtracti  n 
and  multiplication  problems  propounded,  one  woul  I  suppose  our 
children  wore  all  destined  to  become  millionaires.  There  arc 
rarely  any  calculations  which  involve  less  than  a  hundred  thousand. 

Following  ir.  .'alkcr's  lead,  Albert  r.  Boyden  of  the 
Bridgwater*  Massachusetts,  normal  School,  in  1894  revise.!  his 
course  of  study  and  the  methods  of  instruction.   lie  says,  "Arith- 
metic is  too  often  taken  in  a  merely  mechanical  way,  the  pupils 

working  by  rule  with  much  cypher!  ig  and  little  thinking 

study  of  arithmetic  must  be  enriched our  pupil:;  shou 

it  in  less  time  than  is  usually  given  to  the  subject,  an 

the  power  to  think  for  themselves.   How  shall  this  be  accomplished? 

Dy  better  teacliing,  by  better  arrangements  of  subjects by  use 

of  smaller  numbers,  less  figure  work,  b^ solving  such 


-53- 


pr ob lens  as  occur  in  actual  .li£fi. 

In  1902,  President  William  R.  Harper  of  the  Unirer- 

sity  of  Chicago  proposed  a  scheme  for  saving  two  years'  tine  in 
the  completion  of  a  college  course.   His  suggestion  was  a  six- 
year  elementary  course.   Dr.  Jolui  Dewey,  in  disc        is  prob- 
lem says,  "The  proper  aim  of  elementary  tuition  is  to  organize 
the  instincts  and  impulses  of  children  into         interests 
and  tools  Six  years  ought  to  be  enough  to  accomplish 

o 

task.        Out   of   this   ei'fort   to  e  vital 

.uestion,   "'.h  »th  knowtn 

a 
Dr.  Frank  Uc        live  red  an  s       i  i  arithmetic 

before  the  Department  of  Superintendence  of  the  national  Educa- 
tional Association  in  1904  which  stirred  educators  to  furthei 
terect  in  essentials.   He  says  the  content  of  studies  should  be 
determined,  fir;;t,  by  social  needs,  second,  by  the  child's  ability 
to  comprehend.   He  decries  the  "harmonious  development  of  the  facul- 
ties" theory  and  the  puzzle  problem  that  puzzles  the  teacher.   He 
says,  "Life  is  too  full  of  large  specific  aims  to  be  attained  to 
allow  for  work  that  has  no  1  le  object."   He  rejects: 

1.1      ,  "Ibort  C:   Education  14:  page     .     tth  1894. 
2.   Dewey,  John:   Scho         .  "  nuary  1903. 

.  Ions  arc        Le  in  the 

pre.;en\,  :ourso  of  Study  and  What  should  be  the  Basis  for  U 

Same?  National  Edu 

1904. 


-  - 


1.  liatevor  cannot  bo  shown  to  have  a  plain  relation 

to  some  re  1  need  of  life... 

2,  Whatever  is  not  reasonably  within  the  chil  ' 

comprehension. 
3«  Whatever  is  unlikely  to  appeal  to  his  interests* * . 
4.  Whatever  to  ics  and  details  are  so  isolated  or  ir- 
relevant that  they  fail  to  be  a  part  of  any 
series  or  chain  of  ideas... 
These  principles  Dr.  Mc Hurry  pointedly  applied  to 
arithmetic,  and  his  work  became  the  basis  for  much  Invest igati  n 
and  study. 

Guy  lf«  Wilson  and  his  teachers  at  Connersville,  Indiana 
in  1909,  obtained  from  the  business  community  an  o  i  lion  on  en- 
ants  and  inclusions  in  arithmetic. 

The  business  community,  through  their  merchants,  bank- 
ers and  factory  superintendents,  expressed  themselves  In  favor  of 
more  attention  in  the  ublic  school:;  to  the  following  to  ics: 

1.  Saving  and  loaning  money. 

2.  Itortga 

3.  I.Todern  "  thods. 

4.  Building  and   loan  associ 

5.  Keeping  simple  accounts. 
C .  Inv  .oy . 

7.     Bonds  as  Investments* 
G.  as  investments. 

9.     Harks  of  a  good  Investment.      (It  ted  that 

the  gct-rich-qaicl:  concerns  fleece 
Lo   out   of  C60, 0 J), 000  a  ye   r.) 
10.     Taxes,   le.i      ,  Ltures, 


-55- 


11.  Profits  in  different  lines  of  business.     ^ 

12.  Life  insurance  as  protection  and  investment. 

In  1911,  a  course  of  study  based  upon  those  findiu 
was  issued.  This  Connersville  Course  became  the  subject  of  s' 
and  criticism  by  Dr.  Jessup  in  the  University  of  Iowa,  and  by  Dr. 
Coffman  at  the  University  of  Illinois.  They  conceived  the  idea  of 
continuing  the  study  of  inclusions  and  enrichment  through  the  sup- 
erintendents of  all  cities  of  the  united  States  with  a  population 
of  4,00C  and  over. 

Through  the  reports  made  by  the  school  superintendents, 
Dr.  Jessup  recommended  that  more  attention  be  gi^en  to  the  follow- 
ing topics : 

1.  Addition. 

2.  Subtraction. 

3.  Multiplication. 

4.  Division  of  whole  numbers  and  fractions. 

5.  Saving  money. 

6.  Public  utilities. 

7.  Public  expenditures. 

8.  Insurance. 

9.  Taxes. 

10.  Percent 

11.  Profit. 

12.  Build ing  and  Loan. 

13.  Investments. 

14.  Interest. 

15.  Banking. 

16.  Borrowing. 

1.   Wilson,  8.  I'.:   Survey  of  Social  and  Business  Use  of  Arithme- 
tic, Sixteenth  Yearbook,  page  128.  (Also  In  1916  reportof 
Comrittee  on  Elimination  of  Subject  Matter,  Iowa  State 
Teachers  Association,  and  In  A  Survey  of  the  Social  and 
Business  Usage  of  Arithmetic,  Ph.D.  Thesis,  1919.) 


•56- 


17.   Levies.  ± 

10.   Stocks  and  Bonds. 

These  same  superintendents,  Dr.  Je3sup  states,  expressed  them- 
selves as  overwhelmingly  in  favor  of  jiving  special  attention  to 
the  fundamentals  of  addition,  subtraction,  multiplication,  divis- 
ion, and  to  fractions. 

In  1915-1916  Professor  ?/alter  S.  Monroe  made  an  investi- 
gation of  the  economy  of  time  in  arithmetic*  Taking  as  the  aim  in 
the  teaching  of  arithmetic  the  equipping  of  the  pupil  (1)  with  the 
knowledge  of  facts,  principles,  and  relationships  existing  between 
quantities  in  the  solution  of  practical  problems,  and  (2)  with  the 
skills  which  are  necessary  to  perform  these  operations,  he  endea- 
vored to  determine  what  problems  are  practical. 

His  major  purpose  in  this  study  was  to  secure  lists  of 
arithmetical  problems  which  arise  in  human  activities,  and  which 
possess  that  degree  of  utilitarian  or  socialising  value  \ 
justifies  their  being  designated  as  minimal  essentials  of  purpose. 
These  he  rroups  under: 

1.  Occupational  activities. 

2.  Activities  of  the  home. 

3.  Personal  activities. 

4.  Activities  of  school  children. 

Dr.  Calvin  N.  Kendall  and  Dr.  George  A.  Ilrick,  in  ] 

1.  Jessup  and  Goffman:   Supervision  of  Arithmetic,  Attitude  of 
Superintendents,  pa.-e  15.   (Also  Fourteenth  Yoarbo 

2.  Monroe,  V.altor  S.:   Economy  of  Time  in  Arithmetic.  Sixte* 
Yearbook,  page  111. 


aftor  a  commendable  narrowing  of  the  bounds  of  study  by  elimin- 
ating a  mass  of  useless  material,  established  the* following  as 

the  le  itimate  field  of  elementary  mathematics. 

I.  Counting  numbers. 

II.  Re  ad  J ng  numbe  r s . 

1.  Integers  -  Arabic  and  Roman. 

2.  Common  Fractions. 

3.  Decimal  Fractions. 

4.  Denominate  Numbers. 

III.  Writing  numbers. 

1.  Integers  -  Arabic  and  Roman. 

2.  Common  Fractions. 

3.  Decimal  Fractions. 

4 .  Denominate  Numbers . 

IV.  The  Processes. 

1.  Addition.  (a)  Integers. 

2.  Subtraction.       (b)  Common  Fractions. 

3.  Multiplication  of   (c)  Decimal  Fraction  to 

4.  Division.  three  places. 

V.  Percentage  applications. 

1.  Trade  or  Commercial  Discount. 

2.  Profit  or  Loss. 

3.  Commission. 

4.  Simple  Interest. 

VI.  The  following  subjects  should  be  treated  largely  for 
information  purposes: 

1.  Taxes. 

2.  Insurance. 

3.  Stocks. 

4 .  Bonds . 

5.  Bank  Discount. 
Compound  Interest. 

VII.  Denominate  Numbers  in  useful  problems  of  community 
value . 


Kendall  and  Mirick:   How  to  Teach  the  Fundamental  Subjects 
1915. 


■.  • 


VIII.  Geometry  in  so  far  as  it  is  required  in  mensuration 
and  In  making  and  reading  working  drawings  in  shop 
work. 


IX.  Algebra  in  so  far  as  the  use  of  letters  is  required  • 
In  simple  formulas  in  mensuration  and  in  simple  prob- 
lems solved  by  the  equation  method. 

Minimum  essentials  by  David  Eugene  Smith,  as  found  in  a 

chapter  on  arithmetic  in  Teaching  Elementary  School  Subjects ,  by 

Rapeer  and  others  are  as  follows : 


(V.'ork  for  the  first,  second . 
Miss  Worden.) 


third,  and  fourth  grades  is  by 


2»8,  5's,  10's, 


1st  Grade:  (1)  Count  to  100  by  1' 
(2)  The  simple  combinations  to  10  or  12.  (3)  Roman  numerals  i 
seen  on  clock  face.  (4)  l/2,  1/4  In  concrete  way.  (5)  Foot, 
inch,  yard. 


2nd 


>rade : 


(1)  1,000 
10's. 


-  reading,  writing.   (2)  Counting  by 
2's,  3*s,  4's,  9's,  10's.   (3)  Remainder  of  45  combinations.   (4) 
Coin  should  be  recognized  -  $ ,  fL.      (5)  l/2,  l/3,  l/4,  l/8  applied. 
(6)  Multiplication  tables  to  about  5  x  10.   (7)  Addition  of  two- 
figure  numbers  not  involving  "carrying",  and  the  subtraction  of 
such  numbers.   (8)  The  square  and  circle. 

3rd  C- ratio:   (1)  Multiplication  tables  completed.   (2)  Sepa- 
rate numbers  Into  their  prime  factors  and  learn  the  simple  factor* 
(3)  Division  by  ono  digit,  using  Ion;  division  in  the  latter  part 
of  the  year.   (4)  Problem  interpretation. 

4th  Grade:   (1)  Linear  measure.   (2)  Volume.   (3)  Addition 
and  subtraction  of  simple  fractions.   (4)  Simple  decimals. 

5th  Irade:   (Recommended  by  David  Eugene  Smith)   (1)  Review 
four  fundamentals  with  whole  numbers.   (2)  Continue  simple  frac- 
tions, stressing  multiplication  of  fractions.   (3)  Compound  num- 
bers -  "Happily  this  is  becomin.:  less  prominent."   (4)  Decimal 
fractions  are  usual]      a  up. 


6th  Grade:   (1)  Decimal  fractions.   (2)  The  elements  of  per- 


cent' -c 


L.M. 
1.  Rapeer/and  Others:   Teaching  Elementary  School  Subjects 
1917.   Chapters  9  and  10* 


-59- 


7th  Grade :   (1)  The  work  of  this  grade  is  civics,  economics, 
or  sociolof;y,  not  mathematics .   (2)  Interest  (some  mathematics). 
(3)  In  other  countries :   a.  Intuitional  geometry,   b.  Simple  lin- 
ear equation  in  one  unknown,   c.  Graphs,  d.  Factoring.  (*■•  may- 
hope  for  this") 

8th  Grade :   (1)  Same  as  7th.   (2)  Dramatize  the  civics. 

Summary  of  Minimum  Essentials 
by  David.  Eugene  Smith. 

1.  Addition      ) 

2.  Subtraction   )   Whole  Numbers 

3.  Multiplication) 

4.  Division      ) 

5.  Addition      )   Of  decimal  fractions  as  shown 

6.  Subtraction   )    in  the  case  of  U.S.  money. 

7.  The  ability  to  find  a  fractional  part  of  a  number. 

8.  Finding  of  some  percent  of  number. 

9.  How  to  multiply  and  divide  a  mixed  number  (£,  tf)   by 

a  whole  number. 

George  Herbert  Betts  in  his  C lass-Room  Method  and  Man- 
itt  published  in  1917,  says,  "The  main  purpose  in  arithmetic 
is  concrete,  direct,  practical,  applied.   It  is  the  business  of 
Arithmetic  to  enable  one  to  do  the  ordinary  numbering  and  computing 
required  in  the  common  economic  and  social  relations.  The  know- 
ledge required  should  be : 

1.  How  to  count  objects  of  all  kinds.  How  to  count  by  nam- 
ing numbers  only.   How  to  count  by  twos ,  threes,  etc. 

2.  How  to  read  and  write  numbers  of  ten  to  twelvo  figures. 

3.  The  tables  and  processes  involved  in  addition,  subtrac- 
tion, multiplication  and  division  of  whole  numbers. 

1.  BetLs,  George  Herbert:   Classroom  Methods  and  Management.  1917. 
Pages  218-219. 


•60- 


4.  Common   fractions,  and  their  addition,  subtraction,  mul- 
tiplication, and  division  with  the  use  of  such  denominators  as 

are  con  only  used  in  business.  A  similar  knowledge  of  decimals  in- 
volving; up  to  three  places . 

5.  Th6  common  tables  and  measures  employed  in  the  ordinary 
life  routine  of  the  average  man  or  woman.  These  are:  measures  of 
length,  angle,  surface,  volume  and  capacity,  quantity,  weight, 
time,  monoy,  value. 

6.  Our  monetary  system,  denominations,  and  the  various  busi- 
ness practises  involving  the  use  of  checks,  drafts,  notes,  mort- 
gages, etc. 

7.  Percentage,  and  its  simpler  applications  to  practical 
business  uses. 

8.  Simple  mensuration,  applied  tc  lines,  angles,  surfaces, 
volumes . 


Attitudes  to  be  developed: 

1.  A  tendency  not  to  be  satisfied  with  guessing  or  approxi- 
mation, but  to  insist  on  finding  out  through  the  use  of  figures 
on  all  essential  matters  involving  numerical  values. 

2.  Standards  of  business  accuracy  that  will  result  in  the 
keeping  of  an  accurate  account  of  all  personal  or  household  re- 
ceipts and  expenditures.  This  will  make  possible  a  proper  adjust- 
ment of  expenditure  to  income,  and  also  a  right  balance  among  the 
different  objects  for  vrhich  money  is  spent. 

3.  Unwillingness  to  rely  on  general  estimates  or  rough  ap- 
proximations with  reference  to  projects  planned,  as  improving  a 
home,  or  a  farm,  taking  a  trip,  investing  in  an  automobile,  etc. 

4.  Insistence  on  detailed  and  accurately  kept  records  of 
profits  or  losses  from  the  different  enterprises  of  farm,  shop  cr 
business . 

5.  The  development  of  such  a  sense  of  values  and  the  inev- 
itable lo^ic  of  figures  as  will  render  one  proof  against  the  -et- 
rich-quick  schemes  planned  by  unscrupulous  promoters  to  catch 
those  who  through  ignorance  of  business  believe  wealth  to  be  at- 
tained by  some  kind  of  magic. 

6.  A  sense  of  pleasure  and  satisfaction  in  the  use  of  fig- 
ures and  in  the  certainty  which  comos  from  their  wise  application 


■61- 


to  one's  affairs. 

John  Charles  Stone  in  his  Teaching  of  Arithmetic,  1918, 
makes  the  aim  of  arithmetic  practical,  and  outlines  the  essen- 
tials as  follows: 

1.  Efficiency  in  computation. 

2.  A  social  insight  into  business  and  industrial  practices 
that  will  enable  one  to  interpret  references  to  such  practices 
met  in  general  reading  or  in  social  and  business  intercourse. 

3.  Power  to  express  and  to  interpret  the  numerical  expres- 
sions of  the  quantitative  relations  that  come  within  our  exper- 
iences. 

4.  The  habit  of  seeing  such  relations,  particularly  those 
that  are  vital  to  our  welfare.1 

Dr.  Junius  P.  Meriam  of  the  University  of  Missouri,  per- 
haps more  than  any  other  educator,  has  stamped  the  traditional 
arithmetic  as  non-essential.  He  says  that  the  best  way  to  teach 
arithmetic  is  not  to  teach  it  at  all;  that  there  should  be  no 
re  n.ilar  class  periods  and  no  regular  texts;  that  arithmetic  is  a 
school  subject  presented  to  keep  the  child  occupied,  to  keep  him 
from  worse  behavior,  without  considering  the  outcome  of  the  occu- 
pation; that  the  only  arithmetic  worth  bothering  with  is  that 
which  the  child  comes  face  to  face  with  in  life;  that,  however, 
boys  and  rrirls  should,  and  will,  have  considerable  to  do  with 
arithmetic  as  they  experience  quantities  and  measurements  as  they 
help  to  do  those  things  about  them,  and  through  these  will  learn 
to  handle  arithmetic  processes.  He  further  recommends  that  we 

1.   Stone,  John  C:   reaching  of  Arithmetic.  1918. 


•62- 


construct  a  school  course  in  terms  of  normal  activities. 

In  his  Child  life  and  the  Curricula2  Dr.  Meriam  crit- 
icizes unfavorably  the  courses  of  study  in  freneral  use,  all  text- 
books in  part  or  in  whole,  and  the  methods  of  presentation.  He 
says:   Arithmetic  is  a  cross  section  of  a  great  variety  of  exper- 
iences in  the  quantitative  level."   "Arithmetical  abilities  can 
be  measured  by  following  pupils  Into  stores,  shops,  factories  and 

other  places  of  employment  and  there  taking  into  account  the  arith- 

4 
metical  calculations  made  as  part  of  their  work."   "School  arith- 
metic is  strictly  a  form  subject.   It  has  not  yet  approached  the 
study  of  quantitative  aspects  of  our  invironments  and  our  real 
adjustments." 

Dr.  Meriam' s  attitude  toward  arithmetic  in  the  elemen- 
tary school  curriculum  reminds  one  of  that  justly  famous  treatise 
upon  "Snakes  in  Ireland",  the  preface  of  which  contains  the  inci- 
dental observation  that  there  are  no  snakes  in  Ireland. 

Dr.  Edward  Lee  Thorndike  states  his  attitude  toward 

essentials  in  arithmetic  in  this  letter: 

"You  will  find  my  opinion  concerning  what  should  be  In 
and  what  should  be  left  out  of  the  elementary  school  course  in 
mathematics  worked  out  fully  in  the  Thorndike  Arithmetic.  Every  - 

1.  Personal  interview  with  the  writer,  January,  1921. 

2.  Meriam,  J. P.:   Child  Life  and  the  Curriculum.   1920. 

3.  Ibid    page  419. 

4.  Ibid    page  467. 

5.  Ibid    page  286. 


-63- 


tMng  *n  those  except  the  few  exercises  marked  "optional"  up 
to  page  248  of  Book  III  should,  in  my  opinion,  be  left  in, 
together  with  a  selection  from  the  material  on  pages  249  to 
286,  as  stated  on  page  249. 

Yours  truly, 

E.  L.  Thorndike."1 

In  Thorndike's  newest  book,  New  Methods  in  Arithmetic, 

1921,  a  book  built  entirely  upon  the  textbook  material  in  the 

2 
three  volumes  of  the  Thorndike  Arithmetics  (now  adopted  as  the 

State  Text  in  California)  he  advocates  useful  computations  as  op- 
posed to  indiscriminate  ones,  facility  and  absolute  accuracy  with 
small  numbers;  genuine  problems;  arithmetic  for  life.  He  urges 
that  the  older  methods  be  discarded  and  that  the  newer  arithme- 
tic, founded  upon  common  sense,  common  requirements,  common  needs 
be  given  trial,  and  1s  of  the  belief  that  this  will  win  assent  and 
confidence  on  merit. 

Prom  this  examination  of  the  works  of  these  recognized 
leaders  in  thought,  one  seems  justified  in  the  conclusion  that  the 
modern  aim  in  arithmetic  is  a  practical  one,  and  that  throu-h  re- 
search, teachers,  superintendents,  and  reco/mized  educators  are 
endeavoring  to  find  out  what  are  the  clear-cut  first  essentials, 
and  what  shall  constitute  a  minimal  course  of  ;;tudy  in  arithmetic. 

1.  Extract  from  personal  letter  received  by  Chairman  of  Commit- 
tee on  Minimum  Essentials  in  Arithmetic,  Southern  Branch  Univer- 
sity of  California.  1920. 

2.  See  appendix. 


PART 


TWO 


THE 
INVESTIGATION 


•64- 


CHAPTER  I 

SOURCES  AND  METHODS  OF  COLLECTING  THE  DATA. 

The  initial  impetus  for  this  investigation  was  the 
common  work  of  a  Committee  on  Minimum  Essentials  in  Arithmetic  of 
which  the  writer  is  secretary.   This  committee  was  apoolnted  and 

began  its  work  on  February  9,  1918.  A  group  of  Superintendents 

2 
of  Southern  California  City  Schools ,   and  the  President  of  the  Los 

Angeles  State  Normal  School  (now  the  Southern  Branch  of  the  Uni- 
versity of  California)  banded  themselves  together  for  the  better- 
ment of  elementary  school  education. 

"Our  object",  said  their  chairman,  "is  to  attempt, 
through  the  labors  of  a  series  of  carefully  selected  committees, 
to  clearly  define  the  purpose  which  should  regulate  the  teaching 
of  each  of  the  several  elementary  school  studies ,  and  in  accord- 
ance with  that  purpose  reduce  each  of  these  studies  to  its  lowest 
terms  by  eliminating  all  lessons  and  parts  of  lessons  which  do  not 
specifically  contribute  to  that  purpose,  and  to  3tudy  the  best 

1.  Committee: 

Myrtle  Collier,  Chairman,  Southern  Branch, University  of  Calif. 
Katherine  Spiers,  Secretary,    "  " 

Dr.  A.  W.  Plummer,  Los  Angeles.    Bertha  R.Hunt,  Santa  Monica 
Dr.  n.  H.  Snyder,  .    Ruth  Smart,  Long  Beach. 

Dr.  A.  H.  Sutherland,  Los  "    .    Rufus  Mead,  Pasadena. 
Berthile  Barclay,  Santa  Ana.       Frances  Brown,  Riverside. 
Ann  Burnam,  Pomona.  Jessie  Wilkinson,  San  Ber- 

nardino. 

2.  Los  Angeles,  Santa  Ana,  Pomona,  Santa  Monica,  Long  Beach, 
Pasadena,  Riverside,  Redlands,  San  Bernardino. 


■65- 


ways  and  means  of  attaining  that  purpose  in  the  teaching  of 
each  subject." 

The  Committee  on  Arithmetic,  which  is  still  operative, 
met  at  intervals  throughout  the  years  1918  and  1919  and  much  val- 
uable work  was  done.  The  v/riter,  during  the  years  1919,  1920  and 
1921,  has  extended  the  study  and  has  compiled  tins  report  which 
follows  closely  the  line  approved  by  the  Superintendents  of  the 
Southern  Cities  and  the  Committee  on  Arithmetic. 

In  collecting  these  data  an  effort  was  made  to  reach  rep- 
resentative groups.  The  data  of  Questionnaires  I  and  II  were  ob- 
tained through  the  cooperation  of  the  students  of  the  Teacher 
Training  Department  of  the  Southern  Branch  of  the  University  of 

California;  superintendents,  principals,  and  teachers  of  Southern, 

p 

Central,  and  Northern  California  schools;   schools  in  Arizona,  and 

in  Alaska.  The  large  city,  the  small  town,  the  rural  district, 

and  the  remote  outpost  of  civilization  are  represented. 

The  data  of  Questionnaire  III  was  collected  through 

personal  interviews  with,  and  letters  sent  to,  lar  e  business  con- 
's 
cerns  in  Los  Angeles. 

1.  Moore,  E.C:   Address,  Los  Angeles  State  Normal  School,  Feb- 
ruary 9,  1918. 

2.  Grateful  acknowledgment  is  made  to  a  group  of  superintendents, 
principals,  and  experienced  teachers,  members  of  a  seminar 
-roup  in  the  Summer  Session  of  the  University  of  California, 
1319,  for  valuable  help  ,7iven  luring  the  years  1919,1920.  With- 
out their  help  this  work  could  not  'ave  been  carried  on. 

3.  The  writer  thanks  Dr.AIV.  Plummer ,  Principal  of  the  Twenty-ninth 
Street  School,  Los  Angeles,  for  permission  to  use  data  col- 
lected by  Mm  and  embodied  in  this  report. 


-66- 


The  problem  throughout  is  a  positive  one.   It  is  to 
determine  what  arithmetic  men  and  women  actually  do  use,  what  op- 
erations are  employed,  what  figuring  is  actually  done,  and  what 
insight  and  skills  are  required  by  business  men  and  women. 

Questionnaire  I  was  sent  to  the  general  public:  business 
and  professional  men  and  women,  merchants,  shop  keepers,  day  la- 
borers, ranchers,  miners,  cattle  men,  the  leisure  class,  the  home 
keeper.  The  time  of  collecting  covers  a  period  of  one  and  one 
half  ye,ars .  The  purpose  of  this  questionnaire  was  to  discover 
the  size  of  the  numbers  used  in  the  four  fundamentals  with  whole 
numbers,  fractions,  and  decimals,  and  the  type  of  arithmetic  used 
in  daily  life. 

Questionnaire  II  was  sent  to  parents  of  pupils  In  the 
upoer  grammar  grades  and  in  high  schools.  The  teachers  of  many 
schools,  and  the  student-teachers  of  the  Training  School  of  the 
Southern  Branch  of  the  University  of  California  collected  these 
data.  The  questions  were  filled  in  at  different  ten-day  inter- 
vals throughout  a  year.  The  purpose  was  to  determine  what  flgur- 
Ing  is  actually  done  by  people  from  day  to  day,  and  what  arith- 
metical topics  are  in  daily  use. 

The  data  of  Questionnaire  III  are  compiled  from  the  re- 

1.  This  questionnaire  follows  the  line  of  those  of  Dr.  Jes3up 
and  Dr.  Coffman  In  1913. 


-67- 


plies  of  such  prominent  Los  Angeles  business  concerns  as  the 
following:   Lewellyn  Iron  Works;  Santa  Fe  Railway  Company;  Civil 
Service  Commission;  Los  Angeles  Creamery  Company;  Rivers  Brothers, 
Wholesale  Produce  Company;  H.  JeWne  Company,  Grocers;  A  Ham! n 
and  Sons,  Merchants;  R.  L.  Craig  and  Company,  Importers  and  Whole- 
sale Grocers;  Hauser  Packing  Company;  Los  Angeles  Planing  Mill 
Company;  Salt  Lake  Route;  Goodrich  Rubber  Company;  Cudahy  Packing 
Company;  Kahn-3eck  Company;  Bishop  and  Company;  S perry  Flour  Com- 
pany; Los  Angeles  Ice  and  Cold  Storage  Company;  Newmark  Brothers; 
Howard  Brokerage  Company,  Farm  Products;  Globe  Grain  and  Milling 
Company;  Los  Angeles  Public  Library;  Reynold  E.  Blight,  Certified 
Public  Accountant,  others. 

The  purpose  of  th^  s  questionnaire  was  to  determine 
business  needs  in  arithmetic,  to  invite  criticism  of  the  school 
product,  and  to  ask  for  re  commend 'it  ion  toward  improvement  in  the 
school  course  of  study. 


■68- 


Questionnaire  I. 
The  tables  and  charts  on  pages  70  to84  inclusive, 
show,  in  tabular  and  graphic  form,  the  results  obtained  from 
the  questionnaire  to  the  general  public. 

Questionnaire  II. 
The  tables  and  chart  on  pages  87  to  90  inclusive, 
show,  in  tabular  and  graphic  form,  the  results  obtained  from 
the  questionnaire  to  parents. 

Questionnaire  III. 
The  totals  and  charts  on  pages  93  to  95  inclusive, 
show  the  judgment  of  a  group  of  business  men  of  Los  Angeles. 


•69- 


CHAPTER  II 

QUESTIONNAIRE  I. 

(1500  printed  or  mimeographed  copies  3ent  out. 
1136  replies.) 

To  the  Public: 

The  purpose  of  this  questionnaire  is  to  find  out  how 

much  arithmetic  is  used  in  every-day  life.  Do  not  state  what  you 

are  able  to  use  but  what  you  actually  do  use.   State  the  problem 

used  in  questions  1  to  10  inclusive.  Do  not  sign  your  name. 

1.  Please  state  your  occupation 

Answer  question  by  underlining  The  numbers . 

2.  Do  you  pe  r  s  onaTlynave  occasion  to  <?dd  columns  of  2,  3,  4,  5_, 
6,  or  more  numbers  in  height? 

3.  D"o  you  personally  have  occasion  to  add  columns  of  2_$   3_,  4,  5_, 
6,  or  more  figures  in  width? 

4.  Do  you  personally  have  occasion  to  multiply  numbers  of  £,  3_, 
4,  BT~6,  or  more  figures? 


you  personally  have  occasion  to  multiply  by_  numbers  of  2_, 
47~~5 ,  6 ,  or  more  figures? 


5.  Do 

6.  Do  you  personally  have  occasion  to  divide  numbers  of  2_,  3_,  4_, 
5,  6_,  or  more  figures? 

7.  Do  you  personally  have  occasion  to  divide  by_  numbers  of  2,  3, 
4,  5,   6,  or  more  firures? 

8.  How  many  of  the  following  fractions  do  you  personally  have  oc- 
casion to  use:  halves ,  thirds ,  fourths ,  f if tns ,  s ixths , 
sevenths ,  eighths,  ninth's,  tenths,  twelfths,  sixteenths? 

9.  Do  you  personally  have  occasion  to  use  decimals  of  2_,  3_,  £, 
or  more  places? 

10.  Do  you  personally  have  occasion  tf  compute  simple  interest? 

11.  Do  yo\i  yournelf  compute  percentage, 

(a)  when  you  are  paying  taxes? 

(b)  when  you  are  paying  commission? 

(c)  when  you  are  estimat"  tip,   profit  and  loss?^ 

(J)  when  you  are  shopping? 

(e)  when  you  are  paying  insurance? 


Note:   This  form  was  used  by  the  Committee  on  Arithmetic, 
Southern  California  Cities. 


-70- 


Persons 

Replying 

Question 

2 

Question  3 

. 

2 

3 

4 

5 

6  J  lore 

2 

3 

4 

5 

6 

More 

15 

15 

15 

15 

15 

15  7 

14 

14 

15 

15 

14 

10 

19 

19 

19 

19 

19 

19  10 

19 

19 

18 

19 

19 

10 

37 

37 

37 

37 

37 

36  16 

37 

37 

36 

34 

33 

24 

75 

75 

70 

69 

68 

62  44 

75 

72 

69 

55 

44 

29 

43 

42 

42 

41 

39 

34  13 

41 

41 

39 

34 

40 

10 

54 

53 

51 

49 

42 

36  17 

52 

47 

42 

30 

21 

7 

25 

24 

24 

24 

22 

19  12 

24 

24 

23 

15 

8 

3 

9 

9 

9 

9 

6 

6  1 

9 

9 

9 

7 

6 

0 

23 

23 

23 

23 

15 

13  11 

23 

22 

22 

19 

16 

8 

11 

11 

11 

11 

11 

11  6 

11 

11 

11 

9 

6 

3 

108 

105 

104 

101 

94 

84  50 

102 

101 

89 

67 

48 

28 

Accountants 

Attorneys 

Bankers 

Bookkeepers 
Stenographers 

Contractors 

Day  Laborers  — 

Doctors, 
Dentists  

Designers 

Civil  Engineers 

Foremen 

Farmers 

Housewives 237      194  190  173  157  133  74   220  213  177  86  30  19 

Janitors 21      21   21   20   15   11  5    21   21  21  21  19   4 

Librarians 13       13   13   13   13   13  11    12   10  10   9   9   3 

Machinists 18       18   18   18   18   16  8    13   18  15   9   7   3 

Managers 26       26   26   25   25   25  20    26   25  25  21  19   7 

Merchants 156      152  155  144  134  134  80   148  148  134  119  83  35 

Real  Estate  Men—   11       1    11   11   11   11  7    11   11  11   9   8   5 

Teachers, 

Students 42       42   41   39   39   34  23    40   29  36  20  13   7 

Miscellaneous      193      167  176  165  156  152  77   181  177  162  126  96  49 

TOTAL  1136     1077  1056  1005  936  864  492  1004  1049  964  724  539  264 


-71- 


Persons 
Replying 


Question  4 


Question  5 


2  3  4  5  6  More 

15  15  15  15  14  6 

19  19  19  19  19  13 

33  30  28  21  16  6 


68  66  48  46  37  20 
43  42  41  34  23  6 
47   46  41  24  14    5 


Accountant —  —  15 

Attorneys 19 

Bankers 37 

Bookkeepers, 

Stenographers —  75 

Contractors 43 

Day  Laborers 54 

Doctors, 

Dentists- 25 

Designers- 9 

Civil  Engineers  23 

Foremen 11 

Farmers 108 

Housewives 237 

Janitors 21 

Librarians 13 

Machinists 18 

Managers 26 

Merchants 156 

Real  Estate  Men  11 

Teacher 8, 

Students 42      40    38  33  26   17    8 

Miscellaneous—  193     129   178  155  124  88   54 

TOTAL 1136 


23  22   6  7  7 

9  9   9  5  3 

23  22  20  17  14 

10  10  8  6  4 
104  104  97  66  48 
230  198  126  64  34 

21  21  19  15  8 

12  9   6  6  6 

18  18  18  12  9 

24  24  21  14  10 
148  149  125  85  73 

11  11  11  7  6 


2  3  4  5  6  More 

15  15  14  8   6   4 

18  19  16  16  16   9 

25  18  16  13  10   4 

68  62  43  29  20  11 

39  36  33  19  15   4 

46  38  27  14  10   3 

22  18  11  6   3   1 
9  8  6  3   2   0 

23  22  20  18  14   6 

11  11  8  5   4   3 
95  83  65  38  26  12 

216  164  79  30  17  10 

21  17  7  7   5   1 

12  9  8  7   5   1 
18  16  11  6   6   2 

24  20  15  13   9   2 
142  123  85  GO  46  22 

11  8  5  3   3   0 

36  31  19  8  10   6 

178  165  123  61  42  31 


1077  1031  846  613  450  206   1029  873  611  364  269  132 


■72- 


Persons 

Replying  Question  6  Question  7 

2  3  4  5  6  Ifore  2  3  4  5  C  More 

Accountants 15  13  13  13  13  6  4  15  15  13  6  5  4 

Attorneys 19  19  19  ID  19  17  10  19  19  18  14  14  9 

Bankers 37  23  19  18  13  11  6  21  15  11  4  4  0 

Bookkeepers , 

Stenographers—  75  69  59  51  42  48  8  55  41  30  27  27  10 

Contractors 43  40  39  31  26  16  7  40  35  28  23  20  9 

Day  Laborers 54  45  43  40  24  1G  4  45  37  21  9  7  3 

Doctors , 

Dentists 25  22  20  14  9  9  5  21  17  9  3  3  1 

Designers 9               999650  8  4  4330 

Civil  Enginoors  23  23  22  20  16  15  2  23  23  21  16  15  12 

foremen 11     11  11  11  8  5  3  11  11  8  5  5  3 

Farmers 100  100  99  89  61  48  18  98  77  50  28  19  10 

Housewives 237  203  181  149  60  31  17  144  106  28  11  8  8 

Janitors 21     21  20  15  9  6  3  21  18  8  6  4  2 

Librarians 13      12  9  6  3  3  1  12  9  5  3  1  0 

Uacliinists* 18      17  17  16  11  9  3  17  13  8  5  5  2 

Managers 20      21  20  19  15  14  4  21  18  17  15  14  3 

Iferchants 156  133  111  86  70  51  23  135  114  G7  46  M  9 

Real  Estate  Men  11     11  11  11  8  7  1  11  10  5  3  0  0 

Teachers, 

Students 42      37  38  34  25  20  9  36  29  19  10  10  5 

:ascellanoous—  193  164  157  146  114  68  40  169  143  110  75  50  27 


TOTAL 1136     993  922  805  561  399  166   930  768  491  315  248  117 


-73- 


Persons 

Replying  Question  8 

1/2  1/3  1/4  1/5  1/6  1/7  1/8  l/9  l/lO  V 1 2  l/l6 

Accountans 15  14  8  14  11  8   3  10   3  13  7  6 

Attorneys 19  19  19  19  19  16  15  15  15  16  17  13 

Bankers 37  31  26  32  25  23   7  26   6  20  7  4 

Bookkeepers, 

Stenographers-  75  63  56  61  38  29  18  37  17  34  23  21 

Contractors 43  43  36  41  24  19  11  23  15  27  24  13 

Day  Laborers 54  46  24  39  18  16  12  25  11  22  12  9 

Doctors, 

Dentists 25  25  13  20  10  9   6   10   6  10  5  6 

Designers 9  95  7  42272425 

Civil  Engineers  23  20  14  20  14  11  10   7  10  14  13  13 

Foremen 11  11  10  9  6  6   4   8   4  7  2  2 

Farmers 108  49  40  46  23  14   9  17   9  18  11  10 

Housewives 237  183  128  141  65  50  33  46  23  39  17  12 

Janitors 21  20  15  20  13  13  10   8   6  7  8  4 

Librarians 13  12  5  10  83222710 

Machinists 18  15  10  15  9  9   8  12   8  11  7  12 

Managers 26  25  22  20  16  14   6  16   8  11  6  3 

Merchants 156  125  95  119  73  54  41  68  36  65  45  48 

Real  /.state 11  11  8965033065 

Teachers, 

Students 42  41  35  39  22  18  11  23   8  20  9  6 

Hiscellaneous—  193  162  118  160  49  69  36  103  30  89  75  57 

TOTAL lis!  921  687  841  453  388  2*4  466  222  434  297  249 


-74- 


Accountants— 
Attorney? -— — 
Bankers——— — 

Bookkeepers , 
Stenographers- 
Contractors—— 
Day  Laborers — — 

Doctors, 
Dentists—— 

Designers—— — - 

Civil  Enginoors- 

Forerien— — 

Farmers—— 

Houserdvos— — — 

Janitors 

Librarians 

Uachinists 

Llanager3— — — 

Merchants— 

Real  Estate  Uon- 

Toachors, 
Students—— 

Liisce  lloneouo — 

TOTAL 


Persons 
Replying 

Question  9 

Question  10 

2 

3 

4 

More 

Yes 

No. 

15 

15 

15 

9 

5 

14 

1 

19 

17 

18 

8 

5 

19 

0 

37 

33 

28 

15 

1 

33 

4 

75 

65 

56 

30 

15 

52 

23 

43 

33 

27 

23 

7 

35 

8 

54 

44 

33 

13 

4 

31 

23 

25 

23 

16 

8 

5 

23 

2 

9 

8 

6 

2 

1 

8 

1 

23 

22 

22 

21 

8 

11 

12 

11 

10 

9 

6 

4 

8 

3 

108 

84 

53 

42 

24 

65 

43 

23 

149 

31 

14 

4 

160 

77 

21 

20 

14 

11 

1 

18 

3 

13 

12 

6 

5 

1 

8 

5 

18 

16 

11 

10 

3 

8 

10 

26 

18 

8 

6 

6 

17 

9 

156 

130 

87 

69 

25 

99 

51 

1 

9 

6 

2 

0 

11 

0 

42 

36 

17 

14 

8 

32 

10 

193 

165 

137 

70 

52 

66 

127 

113C 

909 

599 

379 

179 

724 

412 

-75- 


Persons 

Replying  Question  11 

(a)  (b)  (c)  (d)  (e) 

Yes  Ho  Yes  No  Yos  l.'o  Yes  No  Yes  No 

Accountants 15  8  7  14  1  14   1  10   5  13   2 

Attorneys 19  12  7  16  3  10   9  3  16  17   2 

Bankers 37  20  17  25  12  22  15  15  22  20  17 

Bookkeepers, 

Stenographers--- ■  75  50  25  52  23  47  28  52  23  51  24 

Contractors 43  25  18  26  17  30  12  28  15  30  13 

Day  Laborers 54  20  34  21  33  17  37  16  38  54  20 

Dootors, 

Dentists 25  10  15  8  17  10  15  11  14  7  18 

Designers 9  2774554365 

Civil  Engineers-  23  12  11  13  10  10  13  12  11  12  11 

Foremen 11  5665568347 

Farmers 108  58  50  75  33  70  38  60  48  55  53 

Housewives 237  68  151  77  1.60  80  157  112  125  57  180 

Janitors 21  11  10  12  9  13   8  14   7  10  11 

Librarians 13  3  10  3  10  4   9  8   5  4   9 

Machinists 18  998  10  998999 

Managers 26  14  12  18  8  16  10  14  12  18   8 

Merchants 156  75  81  96  60  112  44  88  68  112  44 

Real  Estate  Men-  11  8  3  11  0  10   1  6   5  6   5 

Teachers, 

Students 42  19  23  21  21  23   19  25  17  28  14 

Miscellaneous—  193  95  98  92  101  79  114  85  106  82  111 

TOTAL 1136  542  594  596  538  586  550  578  558  573  563 


-76- 

QUESTION  I. 

Please  st^te  your  occupation. 

Accountants 15 

Attorneys 19 

Bankers   --  37 

Bookkeepers,  Stenographers  -  75 

Contractors 43 

Day  Laborers 54 

Doctors,  Dentists  25 

Designers 9 

Civil  Engineers ■ 23 

Foremen 11 

Farmers,  Ranchers 10b 

Housewives 237 

Janitors 21 

Librarians  — 13 

Machinists 18 

Managers 26 

Merchants 156 

Real  Estate  Men 11 

Teachers,  Students 42 

Miscellaneous1 103 

Total        1136 

1.   Under  miscellaneous  are  grouped  those  occupations  which 
were  reported  1,  2,  3,  4,  or  5  times  only,  and  v/hj  ch  log- 
ically could  not  be  included  under  any  of  the  heads  listed. 
Among  these  were:   actors,  (motion-picture)  barbers,  bakers, 
blacksmiths,  brokers,  cleaners,  confectioners,  dressmakers, 
dance  hall  and  picture  show  managers,  electricians,  firemen, 
hotel  and  apartment  house  keepers,  insurance  agents,  land- 
lords, lawyers,  miners,  ministers,  musicians,  nurses, 
peddlers,  photographers,  policemen,  railway  employees,  re- 
tired business  men,  sailors,  soldiers,  sheriffs,  taile   , 
telephone  operators,  undertakers. 


-77- 

QUESTION  II. 

Totals  and  graphic  representation  of  question  number  II  : 

I_>  you  personally  have  occasion  to  add  columns  of  2,  5,   4,  5,  6,  or  more 

"numbers  in  height?  ~"  ~  _____ 

Total  number  persons  replying,  1136 

1  lumber  using  2  addends 1077 


i 

i  1 1 

1 1 


1056 

1005 

936 

864 

492 


GRAPH. 


c 

X 

5f>                         sefo                                                       /o 

Basis  1136 

2  addends 

3  addends 

.  4  addends 

5  addends 

6  addends 

.lore 

1 

■78- 


tUEMluN  III. 


Totuls  and  graphic  representation  of  question  number  ill. 

Do  you  personally  hare  occasion  to  add  columns  of  2,  3,  4,  5,  6  or  more 

figures  in  v.idth? 


Total  number  persons  replying,  - 
Number  using  2  figures  in  width 

■  3 

■  4 

5 

6 
it       ii 


1136 

lot-; 

1049 
964 
724 
039 
264 


c 

i°               i5  ■>/.              soi»               75«/*            /Oo 

Basis   1136 

1 

2  figures 

mmmml^mmmm 

3  figures 

4  figures 

5  fipures 

7t^^^^^^ 

6  figures 

More 

■■■■MB 

79- 


QUESTION  IIII, 


Totals  rj»d  graphic  represent  ation  of  question  rruribor  ]  III: 

io  you  personally  liave  occasion  to    .ultiply  auribors  of  2,  3,  4,  f.. 

fir 

Total  iiuooer  persons  replying 1136 

Multiplicand  2  figures  1077 

1 

846 



40C 

206 


°P 

Stf"                               5C"fc                                     7S¥°                            100'f' 

Basis      1136 

2   figure  rnultiplicond 

3 

4             "                    « 

5              " 



6 

:.!ore     "                                  wmmmm* 

-80- 


QUESTI01I  V. 


Totals  and  graphic  representation  of  question  number  V: 

Do  you  personally  have  oooaslon  to  multiply  by_  numbers  of  2,  3,  4,  5,  ,or  uore 
figures? 


total  number  persons  reply  ing 

Multiplier  2  figures  

"  3 

S»  4 

'»  5 


B 
more 


.  113 
■1C29 
•  875 

■  611 

.  364 


269 


OBIS. 


,<rfo 

5<TJ0                               s 

0  «/"                                  75  «fo 

\ 

/Oi 

1l 

Basis      1136 

2  figure  multiplier 

3          " 

4        » 

5         "                " 

6           "                  " 

More   " 

-81- 


qulst: 


Totals  and  graphic  representation  of  question  number  VI: 

Do  you  personally  have  .ocaslon  to  divide  numbers  of  29  Z,   4,  5,  Oj  or  ore 

figures? 


Total  number  persons  replying 

Dividend  2  figures  

3 


•  —  113 

993 

922 

805 

661 

399 

I 


GEAKI. 


( 

* 

Ibrfo 

so*/'                     rs  sjo 

/oo 

Basis      1136 

2   figure  dividend 



3 

4        " 

5        " 

6        ■ 

More   "              " 

i' 


-82- 


QUESTIOII  VII, 

Totals  and  graphic  representation  of  question  number  VII: 

Do  y  ou  personally  have  occasion  to  divide  by_  numbers  of  2,  3,  4,  6,  C,   or 

figures? 

Total  number  persons  replying US 

Divisor  2  figures 930 


248 


l 

./«                        .    ■ 

50  i'                             7S>1<> 

ZOO'-JO 

Basis        1136 

2   figure  divisor 

" 



4        '• 

5        " 

| 

6         " 

More    " 

^ ^_ 

•83- 


QOLSTIuH  VIII. 

Totals  and  graphic  representation  of  question  number  VIII. 

How  many  of  the  following  fractions  do  you  personally  have  occasion  to  use: 
halves,  thirds,  fourths,  fifths,  sixths,  sevenths,  ei-hths,  ninths,  tenths, 
twelfths,  sixteenths? 


Total  number  persons  replying  —  1136 
TOTAIS 

Halves 921 

Fourths 841 

Thirds 687 

Eighths 466 

Fifths 453 

Tenths 434 

Sixths  - 38t 

Twelfths 297 

Sixteenths  ~  249 

Sevenths 244 

Ninths 222 


OR'.PH 


Basis  1136 

0  «/»             1  5  rf°                               s 

d» 

TS  *f°                              WO 

Halves 

Fourths 

Thirds 

Eighths 

Fifths 

Tenths 

Sixths 

Twelfths 

Sixteenths 

Sevenths 

Ninths 

■■■■■■■•   - 

TIOH  IX. 

Totals  and  graphic  representation  of  quest in  nunber  IX. 

Do  you  personally  Iiave  occasion  to  itso  docimls  of  2,   3,  4,  or  :ore  places? 

Total  nunber  persons  replying  -  1136. 

GR/  ifl 


DM 

909 

599 

379 

—  179 

Basis  1136 

Decimals  2  pla 

2  places 

3   " 

3  places 

4   ■ 
"    more  " 

4  places 
J.tore  " 

100°/° 


-84- 


vUKETlUN  X. 


Totals  and  graphic  represent  Lion  of  question  number  X. 

Do  you  personally  have  occ  sion  t-o  compute  s  imple  interests 


Number  persons  replying,  1136 

Y©s 724 

No  412 


GRy  PH 


0<rfo 

Cf"                                          /0  0 

Basis   113G 

Yes 

No 

qu^:;tiun  xi. 

Totals  and  graphic  representation  of  question  number- 
Do  you  yourself  compute  percentage:  Number  persons  replyinr,  -  1136 

(a)  When  you  are  paying  taxes?  Yes  —  542  No  —  594 

(b)  "    "   "    "    commission?  Yes  —  598  I  o  —  538 

(c)  '        '  estinatinft  profit  and  losnV    Yes  —  586  !.'o  —  550 

(d)  "          "        "     ehopptnjcl  Yes   -  578  la  --   558 

(e)  "          n        "      pay  in;;  insurance?  Yes  —  573  Ilo  —   563 


GH  PH. 


„/0Buais   113f    2i 


>00°j° 


(a)    Yes, 

o, 

(b)    1 
Mo, 

(c)    Yes, 
■o. 

(d)    Yer, 
No, 

(e)   Yes, 

I<o, 

-85- 


JUDGjMENT. 
After  critically  examining  the  tables  and  graphs  0f 
Questionnaire  I,  and  giving  due  esteem  to  the  problems  reported 
—  all  of  which  deal  with  buying,  selling,  measuring;  paying 
wages,  bills  and  debts;  figuring  averages,  simple  percentage  and 
interest  —  the  following  deductions  were  made. 

Minimum  essentials  with  regard  to  size  of  numbers. 

1.  Addition  -  six  addends,  five  figures  wide. 

2.  Multiplication  -  multiplicand,  five  figures  wide. 

multiplier,  four  figures  wide. 

3.  Division  -  dividend,  five  figures  wide. 

divisor,  three  figures  wide. 

4.  Fractions  -  major  group,  halves,  thirds,  fourths, 

minor  ~roup,  fifths,  eighths,  tenths - 

5.  Decimals  -  three  places. 

Essential  computations. 

1.  Figuring  simple  interest. 

2.  Figuring  percent; 

1.   See  Esrentials  in  Problems,  pages  108-109  t    ',      z   study. 


-86- 


CHAPTER  III . 

QUESTIONNAIRE  II. 

(500  mimeographed  copies  sent  out; 
405  replies. ) 

To  Parents : 

The  purpose  of  this  Investigation  is  to  determine  how 
much  arithmetic  is  used  In  every-day  life. 

Will  you  please  tell  your  child  for  each  of  ten  consec- 
utive days  what  you  have  done  with  number;?  durln~  the  day?  Kind- 
ly have  the  child  place  a  cross  (x)  opnosito  the  topic  used,  and 
note  the  problem. 
Please  state  your  occupation. 

1st  2nd  3rd  4th  5th  6th  7th  8th  9th  10th 
Day  Day  Day  Day  Day  Day  Day  Day  Day  Day 

Addition  of  Fractions 

Multiplication  of  Fractions 

Subtraction  of  Fractions 

Division  of  Fractions  


Cash  Checks ,  or  Bills 


Simple  cash  Accounts,  or 
Family  Expense  Accounts 


Proportion 


Stocks  -  -  Dividends 
>nda 


Banking 


Paying  :n  Part  Payments 


Souare  Measure 


Volume 


Board  Measure 


Drawing  to  Scale 


Graphs 


Land  Measure 


Trado  Discount 


.87- 


""  "7I03HMEE 


*% 


II.  cat^?        ijfl+i     jo     3      U«a    M     ft   flS   -do 

®   ©  3    «T  3      c'o      o       OOfrjO      U     85    I.JJ 

ii< «       «  ft  ft   wa   >    ks  o-p   o  mb  so 


Blacksmiths-- 

Bookkoopers" 

Carpenters—— 
Civil  Engineors 

Creamerymon-- 

Electricians— — 

HouseviTTos— 

Laundrynen  ~ 

Hail  Carriers- 

Mochanics— -- 

Merchants 

Miscellaneous— 
Photographers— 

Plumbers— 

Railroadmen— 

Ranchers— • 

P.eal  Estato  Hen- 
Tailors 

Teachers , Students 

TOTAL 


Number 

of  time 

s  used  in  10 

days. 

4 

1G 

0 

0 

0 

2 

G 

0 

0   G 

19 

95 

56 

11 

0 

0 

0 

0 

0  91 

19 

26 

18 

103 

11 

154 

59 

0 

5  17 

5 

11 

3 

3 

1 

3 

7 

18 

4   2 

5 

4 

0 

0 

1 

0 

0 

0 

0   3 

6 

27 

6 

1 

0 

11 

8 

0 

0  17 

125 

102 

56 

19 

41 

15 

18 

0 

9  30 

3 

24 

0 

0 

0 

10 

10 

0 

0  15 

3 

10 

1 

1 

0 

0 

1 

0 

0   0 

26 

25 

47 

84 

34 

51 

78 

0 

4  37 

41 

138 

121 

49 

3 

45 

12 

0 

0  144 

63 

150 

5G 

71 

61 

79 

40 

7 

50  82 

4 

C 

5 

0 

0 

0 

1 

0 

0  26 

3 

15 

0 

0 

0 

0 

0 

0 

0   0 

6 

13 

4 

0 

0 

0 

0 

0 

0   0 

41 

69 

49 

31 

10 

30 

0 

1 

37  21 

9 

55 

28 

0 

0 

0 

2 

0 

0     1 

5 

28 

15 

15 

10 

0 

12 

1 

0   3 

10 

30 

7 

22 

0 

16 

27 

13 

It  1  1 

-88- 


i  -p    c 

X  o     o 
WCC        -p-p       -h      see      a>  a       "S  ►"'j?  7} 


toe      So        oo       rH«-<ooo     x:  r-i        Q^^-t3         £ 

QUESTIONNAIRE  II  §"&    3  2        £2       ^°"^    «3      °3        S  £©    fc    .*•§      » 

(continued)  gg    g  |       ^       g  g  §   -  g     *  a,        S  "|  S    £    Jg     "S 


Blacksmiths 

Bookkeepers 

Carpenters 

Civil  Engineers — 

Creamery  Hen — — — 

Electricians — 

Housewivea 

Laundryraen 

Mail  Carriers- 

Mechanics 

Merchants— — - — 

Miscellaneous—— 

Photographers 

Plunbers 

Railroad  Men 

Rancher  8 

Real  Estate  Men — 

Tailors 

Teachers, 
Students 

TOTAL 406  1455   908  1082   86  1452  2169  51  82  149 


Number 

of  tine: 

1  use 

id  in 

10  days 

4 

19 

10 

15 

0 

21 

37 

0 

0 

3 

19 

159 

77 

74 

5 

96 

149 

1 

0 

2 

19 

133 

79 

94 

15 

76 

60 

7 

3 

14 

5 

24 

20 

34 

4 

9 

16 

0 

0 

1 

5 

4 

1 

3 

0 

25 

22 

.0 

0 

1 

6 

35 

11 

11 

3 

17 

27 

0 

0 

1 

125 

172 

72 

129 

12 

302 

738 

9 

9 

18 

3 

16 

15 

18 

1 

15 

16 

0 

0 

0 

3 

2 

2 

3 

0 

13 

21 

0 

0 

1 

26 

163 

111 

99 

3 

51 

125 

11 

9 

3 

41 

281 

229 

234 

9 

255 

288 

9 

16 

10 

63 

182 

97 

172 

19 

257 

292 

5 

34 

34 

4 

22 

17 

20 

5 

29 

29 

1 

0 

6 

3 

4 

2 

7 

4 

18 

2 

0 

0 

1 

6 

24 

25 

25 

4 

15 

16 

0 

0 

3 

41 

53 

42 

44 

2 

145 

146 

4 

7 

15 

9 

29 

17 

22 

0 

47 

55 

4 

0 

3 

5 

20 

9 

7 

0 

21 

35 

0 

2 

16 

18 

133 

72 

71 

0 

40 

95 

0 

2 

16 

-09- 

QUESTION  I 
Please  state  your  occupation. 


Blacksmiths 4 

Bookkeepers 19 

Carpenters 19 

Civil  Engineers 5 

Creamerymen 5 

Electricians 6 

Housewives 125 

Laundrymen 3 

Mail  Carriers 3 

Mechanics 26 

Miscellaneous^- --  63 

Merchants 41 

Phot  oc  raphe  rs 4 

Plumbers 3 

Railroad  men 6 

Ranchers 41 

Real  Estate  men 9 

Tailors 5 

Teachers,  Students   18 


Total        405 


1.     Under  miscellaneous   are  grouped   those   occupntiom 

were  reported   only   once.      (See   note   page  76,    —Similar 


(See   note   page  76,    —Similar  group—) 


■90- 


2. 
3. 
4. 
5. 
6. 

7. 

9. 

9. 

10. 

11. 
12. 
13. 
14. 

15. 

17. 
18. 


QUESTIONNAIRE  II 

Cash-Fanily 

j-ocpense  Accounts: 

! 

HS9 

Addition  of 
Fractions 

Cash,  CheclB  or 
Bills: 

14-52 

Hultiplioation  of 

Fractions 

I0&2. 

Subtraction  of 

Frictions 

t0  8 

Banking 

«+i 

Trade  Discount 

515 

.faying  in  Part 
nsnts: 

4  72. 

Board  Measure 

4Z5 
41b 

Square  Measure: 

Drwrlng  to  Seal*: 

zes 

Volume 

i  73-, 

Bonds 

/■»"? 

Land  Jeasure: 



Z< 

Divi.-r  on  of 

84 

Stocks,  Dividends 

82 

Proportion 

5/ 

Graphs 

10 

1 

O           <S          Q 

5    5    fc 

t>     ,o      <1      Q      1      « 
%■•*.*       l»       2        "> 

"          1           '•»           K           ~            -M 

-91- 


JUDGMENT . 

Reference  to  the  tables  and  graph  of  Questionnaire  II 
shows  that  the  schools  aro  not  giving  proper  emphasis.  Too  little 
time  is  given  to  some  topics,  as,  Simple  Cash  Accounts,  or  Family 
Expense  Accounts;  Cash  Check,  or  Bills;  Banking;  and  too  much  Is 
-iven  to  others,  as,  Division  of  Fractions j  Volume;  Proportion; 
Stocks,  Dividends,  and  Bonds;  Square  and  Board  Measures. 

The  returns  from  the  section,  "Problems  used  from  day 

to  day",  show  that  wo  are  failing  to  utilize  a  wealth  of  practical 

2 

material  for  problems     by  following  text  books   solely. 

Arit  lime  tic  used   in  evory-day  life. 

Essential  topics. 

1.  Cash  Accounts,  and  Children's  and  Family  Expense  Accounts. 

2.  Addition  of  Fractions. 

3.  Cash  Checks,  or  Bills. 

4.  Multiplication  of  Fractions. 

5.  Subtraction  of  Fractions. 

6 .  Banking . 

1.  People  often  think  they  aro  dividing  by  a  fraction  when  they 
take  a  fractional  part  of  a  number. 

2.  See  section,  Essentials  In  Problems ,  pages  108-109  ^  this  study. 

3.  Let  the  reader  boar  in  mind  that  these  are  essential  tonics, 
not  the  only  topics  to  be  used. 


-92- 

CIIAPTER  IV 

QUESTIONNAIRE  III. 

(100  letters  or  personal  interviews. 
51  replies  in  whole  or  in  part . ) 

"To  Business  Men  of  Los  Angeles: 

Will  you  please  assist  our  public  school  teachers, 

and  thereby  help  the  boys  and  girls,  by  giving  the  following 

questions  careful  consideration  and  sending  your  reports  to  the 

Committee  on  Arithmetic? 

1.  How  much  arithmetic  should  young  people  know  when  they 
enter  your  employment? 

2.  In  what  arithmetic  work  do  you  find  them  weak  or  unsat- 
isfactory? 

3.  What  suggestions  do  you  make  that  may  assist* in  correcting 
mistakes? 

4.  So  far  as  It  comes  to  your  attention,  what  work  in  arith- 
metic is  being  taught  that  is  of  little  or  no  value  in 
your  business?" 


1.   Note  3,  page  65. 


-93- 


Question  1.   How  much  arithmetic  should  young  people  know  when 
they  enter  your  employment? 

51  Replies     oj>      to-/'     20>/'    st>7'     401>     sof>     (,0?°    70Y'     80Y'    90^'  fix- 

Addition               -  51 

' 

Multiplication          -  50 

Division              ,  -  47 

Subtraction             -  45 

Decimals                -  42 

Fractions              -  40 

Percentage              -  32 

Accuracy               -  24 

Mental  Arithmetic        -  16 

Geometry               -  12 

_ 

Proportion             -  8 

_ 

„       . . 

Profit  and  Loss         -  6 

Discount               -   5 

First  year  algebra       -  2 

■ 


-94- 


Question  2.   In  what  arithmetic  work  do  you  find  them  weak  or 
unsatisfactory? 

40  Replies     cW»   /<W°  «•/•  30°f°    40f    5<W  bo>  70°/°    eof  lot"* 

Accuracy              -40 

Addition             -  36 

Decimals              -  36 

Multiplication        -  30 

Fractions             -  28 

Division -  25 

Short  Cuts           -  21 

Percentage           -  20 

Interest             -  16 

Mental  Arithmetic      -  10 

Analysis              -  9 

Subtraction           -  6 

Thoroughness          -  6 

•95- 


Question    3.     Y/hat  suggestions   do  you  make  that  nay  assist   in   correct- 
ing mi  stakes? 


50   Replies 


)f>       30  •/>       A  Of      50f'         tOf'        70f'         60?' 


thoroughness   in  Fundamental! 


Thoroughness   in  Decimals     -46 


Thoroughness   in  Fractions  -42 
Mental  Arithmetic  -35 


Short  Cuts 


Practical  Problems 


''Teach  the  Why" 


Analysis 


'■'■'eights  and  Measures 


Multiplication 
tables  to  20 


Question  4.  T.o   far  as  it  comes  to  your  attention,  what  work  in  arith- 
metic is  being  taught  that  is  of  little  or  no  value  in  your 
business? '- 


50  Replies 


oy>      /of 


All  except  Fundamentals 


All  valuable 


Algebra 


Higher  Arithmetic 


of-       SOf'        bOf1       70 1-      60  f       lot 


•96- 


JUDGMENT , 


Business  firms  demand  skill.  They  ask  for  accuracy 
and  speed  in  handling  the  fundament/ :&  w^iole  numbers,  fractions, 
decimals.  They  require  thoroughness  and  power  to  interpret  prac- 
tical problems. 


Question  1. 
Essentials:  Report  of  51  Firms 

Addition  (whole  numbers) 100  %   of  Firms 

Multiplication  (whole  numbers) 98  %   "  " 

Division  (whole  numbers) 92  %   "  " 

Subtraction  "  88  %   "  " 

Decimals 82  %   "  ■ 

Fractions 78  %   "  ■ 

Percentage 62  %   "  " 

Question  2. 

Employees  weak,  unskilled:  Report  of  40  Firms 

Accuracy 100  %   of  Firms 

Addition 90  %   "  " 

Decimals 90  /£  "  " 

Multiplication 75  %  n  " 

Fractions  70  « 

Division _ .  .   62  %  "  " 

Short  Cuts 52  fa  "  ■ 

Question  3. 
estlons   toward  improvement   of  teaching:      Report   of  50  Firms . 

Thorou-hness    in  fundamentals 100  %  of  Firms 

"  docinal3 92   %  "     " 

"  fractions --• 84  %  " 

Teach  mental  arithmetic 70  %  "    " 

"   short  cuts 60  %  "    " 

Practical  problor^  54        " 

"Teach  t ho  Why" 50        " 


-97- 


Questi  on  4. 

Valueless  arithmetic;  Report  of  50  Firms 

All  except  fundamentals 46  %   of  Firms 

All  valuable   28  %   "     " 

Algebra 16  %   ■ 

Higher  arithmetic 10  %   "    ■ 


-97- 


SUMMAKY. 


ELIMINATION  OF  TOPICS. 
Through  a  careful  comparison  of  the  data  obtained  by- 
means  of  the  three  questionnaires,  with  the  results,  records  and 
judgments  collected  under  Chapter  III,  "Eliminations  V/hich  Have^ 
Been  Suggested  or  Made,"  the  following  sweeping  reduction  in  sub- 
ject matter  is  recommended: 

Eliminations  to  be  made  in  Elementary  Texts  and  Elementary  Courses 
of  Study.1 

1.  Apothecaries  Weight* 

2.  Troy  Weight. 

3.  Longitude  and  Time. 

4.  Furlong  in  Linear  Measure. 

5.  Hand. 

6.  Dram  in  Avoirdupois  Weight. 

7 .  Surveyors '  Table . 

8.  Fathom. 

9.  All  problems  in  reduction,  ascending  and  descending, 

involving  more  than  two  steps. 

10.  G.  C.  D.  as  a  separate  topic. 

11.  All  initial  common  fractions  except  halves,  thirds,  fourths, 

fifths,  sixths,  eighths,  ninths,  tenths,  twelfths, 
sixteenths,  hundredths,  thousandths.  (Initial  fraction 
is  the  fraction  -~iven  for  the  solution  of  a  problem.) 

12.  All  work  with  L.  CM.  except  of  very  low  denominations. 

(As  a  separate  topic.) 

13.  Complex  fractions. 

14.  Compound  proportion. 

15.  Case:;  in  percentage. 

16.  True  discount. 

17.  Compound  interest,  except  in  si  pie  savings  accounts  as 

re-invested  money. 

18.  Problems  in  partial  payments.  , 

1.   All  of  these  topics  are  found  in  Texts  or  Courses  of  Study 
used  in  public  schools  durinf  the  past  ten  years. 


•98- 


19.  Commission  and  Brokerage  (as  applied  to  stocks  and 

bonds . ) 

20.  Profit  and  Loss  as  a  special  topic. 

21.  Knot. 

22.  Partnership  (as  a  special  topic). 

23.  Cube  Root. 

24.  All  algebra,  except  such  simple  use  of  the  equation  as 

is  directly  helpful  in  arithmetic  and  in  other  sub- 
jects met  with  in  the  school  life  of  the  pupil. 

25.  Brackets,  Braces,  Vincula. 

26.  Cancellation,  as  a  special  topic. 

27.  Finding  the  whole  when  a  fractional  part  is  given. 

28.  Paper  tables. 

29.  Gross  and  Great  GrSss. 

30.  Square  (100  sq .  ft.  used  in  roofing). 

31.  Carpeting,  Lumber  Measuring,  Papering,  Plastering, 

Painting,  as  separate  subjects. 

32.  Surveyors'  Land  Measure. 

33.  Foreign  Money. 

34.  Indirect  problems  in  simple  interest.  (Use  the  equation.) 

35.  Bank  Discount. 

36.  Pyramids,  Cones,  Spheres. 

37.  Metric  System.  (To  be  learned  as  occasion  for  use 

arises. ) 

38.  Initial  decimal  fractions  of  more  than  three  places. 

39.  All  problems  v;hose  content  is  outside  the  experience 

of  the  child. 

40.  Examples  of  this  type:  5^-3x7^-6-4. 

41.  All  improbable  problems. 

42.  All  problems  in  which  the  part  and  its  fractional  equiv- 

alent are  given  to  find  the  whole. 

43.  All  so-called  problems  without  number.  ; 

ESSENTIALS  IN  SUBJECT  MATTER. 
If  the  results  indicated  in  this  study  are  to  be  attain- 
ed, and  if  the  returns  from  the  three  questionnaires,  which  defin- 
itely emphasize  the  judgments  of  experimental  research  workers 
along  a  like  line,  be  considered  valid  basis  for  establishing  the 
business  and  social  usage  of  arithmetj c,  the  following  is  the 
legitimate  field  of  elementary  arithmetic. 

I     ^Cr  '    /Kf'i     1,2. 


-99- 


Requircmcnts : 

A  high  degree  of  accuracy  in  solving  problems  involving: 

1.  Addition,  six  addends,  five  dibits  wide. 

2.  Subtraction,  six  dibits  wide. 

3.  Multiplication,  multiplicand  of  five  digits, 

multiplier  of  four  digits. 

4.  Division,  dividend  of  five  digits, 

divisor  of  four  digits. 

5.  Addition,  subtraction,  and  multiplication  of  these 

fractions:  halves,  thirds,  fourths,  fifths,  eighths, 
tenths • 

6.  Addition,  subtraction,  multiplication,  and  division  of 

decimals  to  threo  places. 

7.  Simple  interest. 

8.  Percentage,  (avoid  the  Indirect  problem.) 

9.  Cash  accounts,  and  children's  and  family  expense 

accounts. 

10 .  Banking . 

11.  Common  measures. 

Schools  have  made  arithmetic  unnecessarily  difficult, 
when,  in  truth,  the  work  should  be  simple  and  easy,  as  an  ability 
to  do  practical  problems  is  all  that  is  required  in  life.   The 
average  individual  needs  to  know  how  to  add,  subtract,  multiply, 
and  divide  whole  numbers  and  decimals;  to  add  and  subtract  simple 
fractions;  to  find  a  fractional  part  of  anything;  to  multiply  a 
whole  number  by  a  fraction,  perhaps  a  friction  by  a  fraction;  to 
find  percentages;  and  to  be  familiar  with  comr.ion  measures.  Our 
pupils  should  be  able  to  master  these  necessary  things,  and  master 
them  thorou -hi;;,  y  the  end  of  the  sixth  year.   Studies  in  elim- 
ination and  retardation  which  have  been  carried  on  in  recent  years 
show  that  a  very  -reat  number  of  children  do  not  complete  the 
ci   t-year  elementary  3Chool  course.   In  view  of    a  f  ct,  it  is 


•100- 


of  vital  importance  that  they  be  tau  rht  the  arithmetical  opera- 
tions required  in  life  as  early  as  possible  in  their  school  career. 
The  two  years'  time  thus  saved  might  with  profit  be  utilized  in 
r-iving  boys  and  girls  more  extensive  acquaintance  with  problems 
connected  with  social,  industrial  and  civic  life.  Where  desired, 
a  unified  course  made  up  of  arithmetic,  algebra,  and  geometry,  all 
worked  out  to  fit  life-needs,  might  follow  the  six-year  arithme- 
tic course* 

In  this  brief  and  partial  survey  of  the  outstanding 
needs  of  the  public  in  arithmetic  and  the  relation  of  these  needs 
to  school  instruction,  many  things  have  been  omitted  which  deserve 
consideration,  and  many  questions  have  been  suggested  to  which  no 
definite  answers  have  been  given.  Much  of  the  material  here  pre- 
sented is  already  familiar  to  the  f orward-lokin1  teacher  who  has 
made  a  study  of  arithmetic.   The  aim  has  not  been  to  exhaust  the 
subject,  but  to  point  out  the  real  issues  and  to  Troup  the  facts 
established  around  a  center  of  progress.   It  takes  'any  people 
working  together  to  map  out  a  program  for  the  teaching  of  any 
subject.   The  elaboration  of  the  following  course  of  study  or  the 
pruning  of  it  as  the  critics  judge  fit  must  be  left  to  those  in- 
terested teachers  who  will  use  their  classrooms  as  trying-grounds, 
who  are  ready  to  study  with  new  interest  and  thoroughness  every 
aspect  of  the  subject,  and  who  are  willing  to  weitfh  all  values 
on  the  scales  of  pttblic  need. 


-102- 


5.  Playing  dominoes,  (a)  Matching. 

(b)  Counting  by  5s. 

6.  Time,  (a)  Making  a  clock  face. 

(b)  Hours,  9,  12,  2,  etc. 

7.  Games*  scoring,  (a)  Bean  bag. 

(b)  Ring  toss. 

( c )  Nine  pins . 

(d)  Hook  it. 

(e)  Gue3sinfT  games. 

(f )  Bui Id in  •  up  numbers,  using 
all  possible  combinations. 

8.  Number  stories. 

Through  such  activities,  the  minimum  number  work  of  the 
pupil  should  be : 

First  Grade  - 

1.  (a)  Count  to  20  concretely. 

(b)  Count  to  20  abstractly.   (Symbols  are  to  be 
used  after  the  numbering  knowledge  has  been 
obtained  by  the  use  of  objects  in  work  and 
play . 

2.  Group  objects  by  2's  and  5's  to  20. 

Count  by  2's  and  5's  to  20.   (Grouping  and  count- 
ing symbolized  with  written  words  and  vith  digits. 

3.  Divide  groups  of  objects  into  2's,  3's,  and  4's 
to  12. 

4.  Use  term  halves  when  groups  of  objects  are  divided 
into  two  equal  parts .   (Not  more  than  12  objects 
to  be  used. ) 

5.  Emphasize  the  relationship  between  quantities  by 
means  of  objects.  (Suggestion:  Relationship  be- 
tween inch  and  foot,  pint  and  quart.) 

6.  Denominate  numbers. 

Measurements  - 
12  inch-  1  ft. 

2  pint-  1  qt. 
12  things  ■  1  doz . 
hi  «  1  nickel. 

10j^    s  1  dime 

2  nickels  ■  1  dime. 

7.  The  addition  of  halves  and  halv 


■103- 


Second  Grade  - 

1.  Review  tho  work  of  first  grade. 

2.  Continue  counting  concretely  by  2*s  and  3's  to 
24;  by  4*s  to  40;  by  5's  and  10  *s  to  50. 

3.  Present  the  thirty-throe  combinations,  concrete- 
ly, whose  suns  are  12  or  less.  Aim  toward  an 
abstract  and  sntomatic  mastering  of  these. 

4.  Division  of  groups  of  objects  into  2fs,  3's, 
4!s,  6fs,  to  24* 

5.  Divide  groups  of  objects  into  2,  3,  4,  6,  equal 
parts.   Maximum  12  objects. 

6.  Use  terms  halver.,  fourths  when  objects  are  di- 
vided into  two  or  four  equal  parts. 

7.  Column  addition,  two  digits  vide.   (Sum  of  each 
column  less  than  10.) 

8.  Column  subtraction,  two  digits  ride.   (Each 
digit  in  the  minuend  to  be  greater  than  the  cor- 
responding digit  in  the  subtrahend. ) 

9.  The  addition  of  fourths  to  fourths.   (In  hand 
work  problems . ) 

10.  Denominate  numbers. 

25,^     ■  1  quarter. 
60  min.  •  1  hour. 
7  days  *  1  week. 

Aim  in  the  Third  and  Fourth  Grades  - 

Since  it  is  in  the  t-  ird  and  fourth  grades  that  habits 
of  accuracy  or  inaccuracy  are  formed  in  the  basic  work  in  mathe- 
matics, the  aim  in  these  grades  should  bo  the  automatic  mastery  of 
the  f orty-f 5 ve  combinations  in  addition,  and  the  corresponding  num- 
ber   fact3  in  subtraction;  the  multiplication  tables  through  10 
x  10,  and  the  corresponding  number  facts  in  division.  In  addition 
to  this  the  child  should  learn  column  addition  and  subtraction  In- 
volving the  adding-ln  and  tho  taking-from  processes;  rmiltiplica- 
tion  with  a  multiplier  of  two  or  three  digits,  and  division  with 
a  divisor  of  two  dibits. 


-104- 


Third  Grade  - 

1.  The  forty-five  combinations  in  addition  and 
subtraction  made1  automatic. 

2.  Column  addition  three  digita  wide,  four  addends. 
(With  the  adding-in  process.) 

3.  Subtraction  with  Lhree  digits  wide.   (With  the 
taking-f rom  process . ) 

4.  Multiplication  tables  -  2's,  3's,  4's,  5'3,  6's, 
10' s,  throTigii  6  times  the  number  multiplied. 

5.  Division,  corres ponding  to  the  combinations  in 
the  multiplication  tables,  and  also  with  remain- 
ders. 

6.  Denominate  numbers. 

3  ft.   -  1  yd. 

4  qt.    -   1  gal. 

50J?       »  one-half  dollar. 

Fourth  Grade  - 

1.  Continue  work  of  third  grade. 

2.  Addition,  four  digits  wide  and  four  addends,  also 
three  digits  wide  and  five  addends. 

3.  Subtraction,  four  digits  vide. 

4.  Complete  the  multiplication  tables  throuh  the 
10's.  (lu  x  10)  Multiplicand  4  dibits  wide, 
multiplier  2  dibits  v.ide.  (Using  dollars  and  c 
cents . ) 

5.  Division:   Dividend  not  more  than  five  digits; 
divisor  not  more  than  two  digits. 

6.  Denominate  numbers. 

16  02.    -  1  ]b. 
10  dimes  =  |1. 
100  cents  s  $1. 

Fifth  Grade  - 

The  aim  in  the  fifth  grade  should  be  the  mastery  of  com- 
mon fractions. 

1.  Continue  drill  on  the  four  fundamentals  with 
whole  numbers . 

2.  Fractions. 

(a)  Continue  fraction  work  of  the  previous 

jes . 

(b)  The  addition  of  fractions  in  the  follow- 
ing order  of  groups : 

Halves  and  halve c . 


-105- 


Halves  and  fourths. 

Halves,  fourths,  eighths. 

Halves,  thirds. 

Halves,  thirds,  sixths. 

Halves,  thirds,  sixths  and  twelfths. 

Halves,  tlirds,  fourths,  sixths  and 

twelfths. 
Fifths  and  tenths. 

(c)  Reduction  of  fractions  when  necessary  in 
addition  and  subtraction. 

(d)  Addition  and  subtraction  of  mixed  num- 
bers.  In  addition,  tv;o  digits  wide, 
three  addends . 

(e)  Multiplication  of  fractions  -  l/2,  1/3, 
2/3,  l/4,  3/4  each  repeated  a  given  num- 
ber of  times.   (Example,  3  x  1/2  -  3/2.) 
1/2,  1/3,  2/3,  1/4,  3/4,  1/10  of  a"  ^roup. 
(Example  2/3  of  18.)   Multiplication  of 
simple  mixed  numbers  (In  playing  store, 

etc . ) 

(f)  Division  of  fractions.   How  many  l/2's, 
l/3fs,  1/4' s,  3/4's,  l/io fs  in  whole  num- 
bers. Example,  how  many  3/4's  In  6,  or 

6  -T-  3/4  =  ?  Develop  throu  "h  the  concrete, 
use  abstractly. 
3.  Denominate  numbers . 

(a)  Review  -  Continue  the  work  of  previous* 
grades . 

(b)  Hew  -  24  hr.  r  1  day. 

Sixth  Grs.de  - 

1.  Drill  on  the  fowr  fundamentals  ?'n  whole  numbers 
and  fractions. 

2.  Decimal  fractions. 

(a)  The  decimal  Idea. 

(b)  The  four  fundamentals  in  decimals,  with 
a  limit  to  tvo  decimal  places  in  initial 
decimals  (Stress  dollars  and  cents). 

3.  Per  cent. 

1/100  .  .01  x  1%.      Nothing  new. 

4.  Find  the  per  cent  of  a  number. 

5.  Find  what  percent  one  number  is  of  another  (18  is 
what  per  cent  of  20?) 

6.  Denominate  numbe/ 

(a)  Review  denominate  numbers  through  prob- 
lem work. 

(b)  New  work  to  be  taken  up  through  problems . 


•106- 


100  lbs.   =1  cwt. 

2000  lbs.   »  1  ton. 

144  sq.in.«  lsq.  ft. 

9  sq.ft.s  1  sq.  yd. 

160  sq.rd.»  1  Acre. 

60  sec .   »  1  minute . 

365  days  ■  1  year. 

12  mo.   »  1  year. 

Seventh,  eighth  and  ninth  grade  mathematics  should  be  a 
unified  course  made  up  of  arithmetic,  algebra,  and  geometry,  and 
presented  in  such  a  way  that  there  will  be  no  definite  break  from 
arithmetic  into  algebra. 

Seventh  grade  - 

1.  Drill  for  speed  and  accuracy  with  whole  numbers, 
fractions  and  decimals. 

2.  The  application  of  numbering  to  real  life  needs: 

(a)  Social. 

(b)  Industrial. 

(c)  Civic. 

3.  Discount  -  a3  per  cent. 

4.  Interest. 

(a)  Money. 

(b)  Investments. 

5.  Commission  -  in  connection  with  vocational  guid- 
ance. 

6.  Taxes  -  local;  in  simple  form  in  connection 
with  civics. 

7.  Personal  accounts. 

8.  Banking; 

(a)  How  to  make  out  a  deposit  slip. 

(b)  How  to  write  a  check. 

(c)  Hon  and  why  to  fill  the  stub. 

(d)  When  a  check  should  be  cashed. 

(e)  How  to  stop  payment  on  a  check. 

(f )  How  to  indorse  a  check. 

(g)  llo.v  to  indorse   a  note. 

(h)  How  to  write  a  negotiable  note. 

(i)  How  to  compute  interest. 

(j)  Importance  and  purpose  of  savings  banks. 

(k)  Importance  and  purposebf  commercial  deposit 

banks. 
(1)  How  to  open  an  account, 
(m)  How  to  secure  a  bank  draft, 
(n)  How  to  use  a  bank  draft. 


•107. 


9.  The  solution  of  the  simple  equation  in  algebra. 

(a)  Definition  of  equation. 

(b)  Making  equations. 

(c)  Solving  equations  by: 

Adding  the  same  amount  to  both  sides. 
Subtracting  the  same  amount  from  both 

3ides. 
Multiplying  both  sides  by  the  same  amount. 
Dividing  both  sides  by  the  same  amount. 

(d)  Use  of  the  parenthesis  in  equations. 

(e)  Positive  and  negative  numbers  in  the 

equation - 

10.  Intuitive  geometry  based  upon  shape,  size  and 
location  of  objects. 

(a)  The  rectangle. 
(x)  Perimeter, 
(y)  Area. 

(b)  Triangle, 
(x)  An;;les. 

(y)  Similar  triangles  -  in  solution  of 

problems  only, 
(z)  Construction  of  similar  triangles. 

11.  Use  of  the  compasses  in  drawing  straight  lines, 
dropping  a  perpendicular  from  a  point  to  a 
line,  erecting  a  perpendicular  at  a  given  point 
in  a  line,  constructing  equal  angles. 

12.  Use  of  the  protractor  in  measuring  angles. 

Eighth  Grade  - 

1.  Farther  development  of  Seventh  grade  work. 

2 .  Graphs . 

3.  The  right  triangle  -  Theorem  of  Pythagoras. 

4.  Relation  of  opposite  angles  and  of  the  angles 
made  by  a  transversal  cutting  parallel  lines. 

5.  Cubical  contents  of  rectangular  prisms. 

6.  Cubical  contents  of  cylinders. 

The  work  in  numbering  must  be  a  unit  from  the  first 
grade  through  the  eighth  (Example:  After  fractions  have  been 
learned  there  is  nothing  new  in  decimals  but  the  for1-!  of  vriting. 
Per  cent  Is  only  a  particular  common  fraction.   l/lOO,  or  a  par- 
ticular decimal,  .01  *  l/lOO  ■  1%.) 

Much  time  will  be  saved  by  this  unity  of  work.   Further- 


•ion- 


more,  the  child  will  be  able  to  master  the  subject  of  aritnmetic 
if  this  simplicity  of  the  subject  is  shown. 

ESSENTIALS  IN  PROBLEMS. 

1.  All  problem  vork  should  be  such  as  will  develop  the 
child's  ability  in  numbering  --  that  is,  the  content 
of  the  problem  must  be  within  the  comprehension  of 
the  child. 

2.  Children  should  make  and  solve  problems  growing  out  of 
their  own  experiences  or  environments,  covering  the  num- 
ber work  as  outlined  for  the  various  grades,  using  the 
suggested  topics  of  activity  as  a  guide. 

3.  In  problem  solving  the  five  steps  are  to  be  learned: 

a.  State  the  problem  clearly,  or  read  it  understand- 
ing ly. 

b.  Pick  out  the  unknown  fact  or  facts. 

c.  Choose  the  form  of  relating  the  known  facts  (add, 
subtract,  multiply,  or  divide)  In  order  to  deter- 
mine the  unknown  fact  or  facts. 

d.  Solve.  (This  involves  the  mechanics  of  the  subject.) 
The  child  should  approximate  his  result  before 
solving.  The  result  should  always  be  checked  by 
the  pupil.1 

The  returns  from  that  section  of  Questionnaires  I  and 
II,  "State  the  problem,"  indicate  that  the  stereotyped  problems 
in  the  majority  of  textbooks  in  use  to-day  are  not  problems  met 
In  life.  The  following  list  is  compiled  from  these  returns,  and 
from  personal  observation  of  the  things  children  really  do. 

Problem  Material:   Real  grocery  bills,  making  chati'o, 
games,  value  of  a  cafeteria  meal,  measurements;  inch,  foot,  yard, 
mile,  (fraction  of),  acre,  dozen,  (fraction  of)  liquid  {pints, 

1.   Prom  "Minimum  Essentials,"  1921.  (to  be  published.) 


■109- 


quarts,  gallons,)  time,  U.  S.  Money,  pound,  ton,  averages,  home 
garden  problems,  errands,  cost  of  hnating  and  li -hting  a  home, 
reading  meters,  cyclometer,  pedometer;  wages,  labor  questions, 
chicken  business,  plotting  gardens  and  courts  for  games,  measuring 
land  and  city  lots,  cost  of  furnishing  a  home,  cost  of  building  a 
garage  —  cement,  lumber,  labor,  paint,  draw  to  scale,  insure, — 
borrowing  and  lending  monoy,  commissions,  insurance,  local  taxes, 
Shopping,  dairy  business,  interest,  bond  coupons,  banking  (every 
phase  met  with  by  depositors),  income  tax,  discount  in  shopning, 
household  accounts,  budgets,  sugar  and  cotton  industries,  frc 
ing,  fire  department,  city  market,  real  estate,  excavations,  road 
and  street  building  and  repairing,  transportation  —  street  car, 

railroad,  ship, —  percentages  in  games ,  races,  field  meets, 

others  provided  they  meet  this  standard: 

"The  good  textbook  and  the  good  teacher  scrutinize  every 
task  they  assign  to  make  sure  that  it  fits  the  pupil  Tor  life. 
They  seek  to  find,  for  every  aritiimetical  principle  and  fact,  the 
real  affairs  to  which  it  aoplies  and  with  which  it  should  be  con- 
nected in  the  pupil's  mi ad* 

1.  Thorndike,  Edward  Lee:  New  Methods  in  Arithmetic.  1921. 


-110- 


BIBLIOGRAPHY 


ARNETT,    L.D. 
AYRES,    LEONARD    P. 

BAGLEJC,   W.    C. 
BALL,K.   And  WEST,M.E. 

BALTIMORE  COUNTY  COURSE 
BARNES,  EARLE 
BETTS,  GEO.  HERBERT 

BOBBITT,  FRANKLIN 
BONSER,  FREDERICK 
BOYDEN,  ALBERT  G. 

BRANFORD,  BENCH ARA 
BRESLICH,  E.  R. 


BROWN,  J.  C. 


BROWN, J. C.  and  COFFMAN, 
BROWNE,  CHARLES  E. 


Counting  and  Adding.  American  Journal  of 
Psychology.  Vol.  16,  pp.  327-336. 

Elimination  of  Unprofitable  Subject  Matter. 
National  Education  Association.  1913. 
pp.  243-244. 

Educational  Values.  Chapter  12.  1911. 

Household  Arts  Arithmetic.  School  Review, 
Vol.  25.   Dec.  1917.  pp.  722-730. 

OF  STUDY.   1919.  pp.  261-329. 

An  Old  Arithmetic.  1655.  Academy  4,  p. 502. 

Classroom  Method  and  Management.   Chapter 
1-9,  and  Chapter  12. 

The  Curriculum.  1918. 

The  Elementary  School  Curriculum.  1920. 

The  Essentials  of  Arithmetic.  Education. 
Vol.  14,  pp.  390-400. 

A  Study  of  Mathematical  Education,  Includ- 
ing the  Teaching  of  Arithmetic.  Oxford, 
Clarendon  Press.  1908.  p.  392. 

Supervised  Study  as  a  Means  of  Providing 
Supplementary  Individual  Instruction  in 
Mathematics.  Thirteenth  Yearbook  of  the 
National  Society  for  the  Study  of  Educa- 
tion. 1914.  pp.  32-73. 

An  Investigation  of  the  Value  of  Drill 
Work  in  the  Fundamentals  of  Arithemtic. 
Journal  of  Educational  Psychology,  Vol.2. 
1911,  pp.  81-88;  Vol.  3,  1912.  pp.  485- 
492  and  pn.  561-570. 

LOTUS  D.   How  to  Teach  Arithmetic.  1914. 

The  Psychology  of  the  Simple  Arithmetical 
Processes.   American  Journal  of  Psychology 
Vol.  17.  Jan.  1906.  pp.  1-37. 


-Ill- 


BULLETIN  OP  THE  ST.VTE  NORMAL  SCHOOL.   Superior,  '.is.,  Oct.  1915. 
CAJORI,  FLORIAN 


CAMERER,  ALICE 

CHARTERS,  I.  W. 
CHASE,  S.  E. 
COLBURN,  WARREN  S. 

COLIINS,  JOSEPH  V. 

COOKE,  FLORA  J. 

COURTIS,  S.A. 
CUBBERLY,  ELWOOD  P. 
CURRICULUM,  THE 

DE'„EY,  JOHN 


DIL   ORTH,    THOMAS 


History  of  Elementary  Mathematics.  1917. 
pp.  215-219. 

What  Should  be  the  Minimum  Information 
About  Banking?  Seventeenth  Yearbook  of  the 
National  Society  for  the  Study  of  Educa- 
tion. 1918.  pp.  18-27. 

Teaching  the  Common  Branches.  1913.  Chap- 
ter 12. 

Waste  In  Arithmetic.  Teachers'  College 
Record.  Vol.  18,  Sept.  1917.  pp.  360-370. 

An  Intellectual  Arithmetic.  1821;  Sequel 
to  Intellectual  Arithmetic;  1828;  An  Ad- 
dress Delivered  in  1830  on  Arithmetic. 
Elementary  School  Teacher.  June  1912. 

The  Superintendent  and  the  Course  in 
Arithmetic.  Educational  Review,  Vol.  27. 
1904.  pp.  83-89. 

Minimum  Requirements  in  Francis  W.  Parker 
School.  Elementary  School  Teacher,  Vol. 
12,  p.  245. 

Standard  Tests  in  Arithmetic. 

The  Changing  Conception  of  Education.  1909, 

Mathematics.  Fifteenth  Yearbook  of  the 
National  Society  for  the  Study  of  Educa- 
tion, 1916.  pp.  65-67. 

Democracy  and  Education.  1916. 

Schools  of  Tomorrow.  1915. 

The  School  and  Society.  1900. 

Shortening  the  Years  of  Elementary  School- 
ing. School  Review,  Vol.  2,  Jan.  1903. 

Schoolmaster's  Assistant.  Published  in 
England  In  1743  -  America  in  1803. 


-112- 


DRUSHEL,  J.  ANDREW 

EDUCATIONAL  RESEARCH. 
EDUCATIONAL  RESEARCH. 

ELIOT,  CHARLES  W. 

FREELAMD,  GEORGE  E. 
pre:-: MAN,  F.  N. 
GAULT,  P.  B. 

GREENWOOD,  J.  H. 

QRIGGS,  A.  0. 

HAGGERTY,  MELVIN  E. 

HALL-CiUEST,  ALFRED  L. 
HART,  WALTER  W. 

HECKERT,  J.   . 


A  Study  of  the  Amount  of  Arithmetic  at 
the  Command  of  the  Higli  School  Graduates. 
Elementary  School  Journal,  Vol.  May,  1917. 
pp.  657-666. 

Los  Angeles  City  School  District.  Bulle- 
tin No.  2.  First  Yearbook.  July,  19 

American  Committee  No.  1.  International 
Commission  on  the  Teaching  of  Mathematics. 
Mathematics  in  the  Elementary  Schools  of 
the  United  States.  Bulletin.  1911.  United 
States  Bureau  of  Education. 

Educational  Reform.  1898. 

The  Concrete  and  Practical  in  Modern  Edu- 
cation. 1913. 

Modern  Elementary  School  Practice.  1919* 

The  Psychology  of  the  Common  Branches.  1916, 

Arithmetical  Progression.  Education,  Vol. 
20,  1900.  pp.  295-297. 

Evolution  of  Arithmetic  in  the  United 
States.  Education,  Vol.  20.  1899.  pp. 
193-295. 


Peda-^ocrv  of  Mathematics 
Seminary,  Vol.  19.  1912, 


Pedagogical 
pp.  359-375. 


Studies  in  Arithmetic.  Indiana  Univer- 
sity Studies.  Vol.  3.  Sept.  1915,  No.  32, 

The  Textbook.  1918. 

Community  Arithmetic.   Elementary  School 
Teacher.  1911.  po.  285-295. 

Cleveland  Survey  Testa  In  Miami  Valley. 
Elementary  School  Journal.  Vol.  18.  p. 
447. 


HELMAN,  J.  D.  and  SHULTES,F  .1 .   A  Study  in  Addition.  Research 
Bulletin  No.  1. 


,  HENRY  B, 


A  Foundational  Study  in  the  Peda~o  ;  of 
Arithmetic.  1914. 


•113- 


INTE  RNATIONAL  COMMISSION  on  the  Teaching  of  Mathematics.   Report 
of  the  American  Commissioners.  United 
States  Bureau  of  Education.  Bulletin.  1912. 
No.  14. 

INTE  NATIONAL  COMMISSION  on  the  Teaching  of  Mathematics.   Mathe- 
matics in  the  Elementary  Schools  of  the 
United  States.   United  States  Bureau  of 
Education.  Bulletin  1911.  pp.  75-100. 

IOWA  STATE  TEACHERS  ASSOCIATION.   Report  of  Comm5tt.ee  on  Elimina- 
tion. 1916. 


JACKSON,  L.L, 


The  Educational  Significance  of  Sixteenth 
Century  Arithmetic.  Teachers1  College 
Record.  1901.  pp.  232. 


JESSUP,  W.  A.  and  COFF  AN,  LOTUS  D, 
1916. 


JESSUP,  WALTER  A. 


Supervision  of  Arithmetic. 
School 


Standards  and  Current  Practices 
and  Society.  July  24,  1915. 

Grade  for  the  Introduction  of  a  Text  in 
Arithmetic.  Elementary  School  Journal 
Vol.  15.  Nov.  1914.  pp.  162-166. 

Eliminations  in  Arithmetic  as  a  Factor 

in  the  Economy  of  Time.  National  Education 

Association  Report,  pp.    209-222. 

Economy  of  Time  in  Arithmetic.  Elementa- 
ry School  Teacher.  Vol.  14,  1914.  on. 
461-476. 


JOHNSON,  GEORGE 
JONES,  OLIVE  M. 
JUDD,  CHARLES  E. 


KENDALL  and  MIRICK 

klappi:r,  PAUL 


Education  by  Plays  and  Games.  1907. 

Teaching  Children  to  Study. 

Measuring  the  Work  of  the  Public  School. 
Cleveland  Foundation  Survey.  1916. 

Introduction  to  Study  of  Education.  His- 
tory of  Mathematics.  1918. 

How  to  Teach  the  F\indamental  Subjects.  1915 

The  Teaching  of  Arithmetic.  1916. 


-114- 


MASSACHUSETTS  BOARD  OF  EDUCATION.  A  Course  of  Study  in  Arithme- 
tic. Boston,  Massachusetts. 


MEAD,   CYRUS  D. 
MERIAte,    Junius    L. 
McLELI -',"'   and  DEWEY 
MCMU;  \y  ,    CHARLES  A. 
McMUb^Y  >    FRANK  M. 


An  Experiment  in  the  Fundamentals. 

Child  Life  and  the  Curriculum.  1920. 

The  Psychology  of  Number.  1895. 

Special  Methods  in  Arithmetic.  1905. 

What  Omissions  are  Advisable  in  the  Pres- 
ent Course  of  Study.   Report  of  Proceedings 
of  National  Educational  Associations.  1904. 

Elementary  School  Standards  -  Attention  to 
Relative  Values.  1913.  pp.  115-119; 
Arithmetic  Recommendations,  p.  167. 

The  Uniform  Minimum  Curriculum  with  Uni- 
form Examinations.  Addresses  and  Proceed- 
ings of  National  Education  Association. 
1913.  pp.  131-148. 

MINNESOTA  TEACHERS  REPORT  ON  ELIMINATION.   1915.  Eulletin  State 

Department  of  Education,  St.  Paul. 


MITCHELL  ,  A.  E. 


monroe,  Walter  scott 


Some  Social  Demands  of  the  Course  of 
Study  in  Arithmetic.   The  Seventeenth 
Yearbook  of  the  National  Society  for  the 
Study  of  Education.   1918.  pp.  7-18. 

Educational  Measures  and  Standards. 
1914-1915. 

Derivation  of  Reasoning  Tests  in  Arithme- 
tic. School  and  Society.  Vol.  8.  Sept. 
7-14;  1918,  pp.  295-299,  324-329. 

Warren  Col burn  on  the  Teaching  of  Arith- 
metic together  with  an  Analysis  of  his 
Arithmetic  Texts.  Elementary  School 
Teacher,  Vol.  12,  po.  463-480. 

Development  of  Arithnetic  as  a  School 
Subject.   U.S.  Bureau  of  ;/iucation,    1- 
letin  1917.  No.  10,  pp.  1-170. 


•115- 


MONROE,  WALTER  SCOTT 


MOORE,  ERNEST  C 


Series  of  Diagnostic  Tests  in  Arithmetic. 
Elementary  School  Journal,  Vol.  19.  April 
1919.  pp.  585-607. 

Analysis  of  Colburn's  Arithmetics.  Ele- 
mentary School  Teacher,  Vol.  13,  1913, 
pp.  239-246. 

Development  of  Arithmetic  Teaching  in 
United  States.  Elementary  School  Teacher. 
Vol.  13,  pp.  17-24. 

Principles  and  Methods  in  Teaching  Arith- 
metic as  Derived  from  Scientific  Investi- 
gation. The  Eighteenth  Yearbook  of  the 
National  Society  for  the  Study  of  Educa- 
tion.  Part  2. 

A  Preliminary  Report  of  an  Investigation 
of  the  Economy  of  Time  in  Arithmetic.  The 
Sixteenth  Yearbook  of  the  National  Society 
for  the  Study  of  Education.  1917.  pp. 
111-127. 

Analysis  of  Oolburn's  Arithmetics  4  and  5. 
Elementary  School  Teacher,  Vol.  13,  pp. 
239-294. 

Does  the  Study  of  Mathematics  Train  the 
Mind  Specifically  or  Universally?  School 
and  Society,  Vol.  7.  1918.  pp.  137-140. 

What  is  Education?  1914- 

What  the  War  Teaches  About  Education.  1919, 


MORRISON,  H.  C 


NA1I    NAL   EDUCAVI   H    ki 


Reconstructed  Mathematics  in  the  '  ' 
School:  the  adaptation  of  Instruction  to  the 
Needs,  InterestB,  and  Capacities  of  Stu- 
dents .   The  Thirteenth  Yearbook  of  the 
Nati  onal  Society  for  the  Study  of  Educa- 
tion, 1314,  pp.  9-32. 

lOCIATION  REPORT.   Hov  a  Course  of  Study 

should  be  Determ'ned.  1914,  pp.  235-243; 
223-235. 


NATIONAL  EDUCATION  ASSOCIATION.   Report  on  the  Correlation  of 
Studies.  1895.  pp.  709-714. 


-116- 


PARKER,  SAMUEL 
PHIL:  IPS,  F.  M. 

PIKE,  NICHOLAS 
PYLE,  W.  H. 

RAPEER  and  OTHERS 

REEVE,  V..  D. 
RICE,  J.  M. 

RUGO,  HAROLD  0. 
SCHORL!  I! G,  R. 

,  DnVID  EUGENE 


SMITH,  ARTHUR  G. 
STAMPER,  ALVA  I . 


History  of  Modern  Elementary  Education. 
1912. 

Value  of  Daily  Drill  in  Arithmetic.  Jour- 
nal of  Educational  Psycholo.-y,  Vol.  4, 
March  1913.  pp.  159-163. 

New  and  Complete  System  of  Arithmetic.  1788. 

Economical  Learning*   Journal  of  Educati  n- 
al  Psychology,  Vol.  4.  1913.  pp.    148-158. 

Teaching  of  Elementary  School  Subjects. 
1917.  Chapter  by  David  Eugene  Smith,  pp. 
207-249. 

Unification  of  Mathematics  in  the  High  School 
School  and  Society.  1916.  pp.  203-212. 

Causes  of  Succes   and  Failure  in  Arithmetic. 
Forum,  34,  pp.  437-452;  281-297. 

Essentials  in  Elementary  Education.  Forum. 
1897.  Vol.  22,  pp.  538-546. 

Statistical  Methods  Applied  to  Education. 
1917. 

Significant  Movements  in  Secondary  Mathe- 
matics. Teachers'  College  Record.  1917. 
pp.    438-457. 

Article  on  Arithmetic  in  Monroe's  Cyclo- 
pedia of  Education.  1911.  Vol.  1,  po.  203- 
207. 

The  Teacing  of  Arithmetic.  1913. 

The  Te-ichin  -  of  Elementary  Math  matios«  1901. 

Mathematics  in  the  Training  for  Citizenship. 
Teachers'  College  Record.  1917.  pp. 211-225. 

Number  Games  and  Number  Rhymes. 

Teaching  of  Arithmetic.  Schoul  Science  and 
Mathematics.  Vol.  12.  1912.  pp.  457-460. 

A  Textbook  on  the  Teaching  of  Arithmetic. 
1913. 


•117- 


STARCH,    D 


STONE,    CLIFF    '.VI  N FIELD 


STONE,    JOHN   CH 


STRACHAN,  JAMES 

SUZZALG,  HENRY 
THORNDYKE,  EDWARD  L. 


TAYLOR,  JOSEPH  S. 
THOMPSON,  THOMAS  E, 
WILSON,  G.  M. 


Transfer  of  Training  In  Arithmetical 
Operations.   Journal  of  Educational  Psy- 
Chology,  Vol.  2,  pp.  306-310. 

Questionnaire  to  the  Business  Men  of  Ind- 
ianapolis . 

Arithmetic  Considered  as  a  Utilitarian 
Study.  Elementary  School  Teacher.  1903. 
pp.  533-542. 

Reasoning  Tests.  Pub.  Teachers'  College, 
H.  Y.  1916. 

Arithmetical  Abilities  and  Some  Factors 
Determining  Them.  1908. 

The  Modernization  of  Ar-iLVimetic.  Journal 
of  Education,  Vol.  78,  1913.  pp.  541-542; 
548-549. 

Ihe  Teaching  of  Arithmetic.  191  . 

Mathematics  -  The  New  Teaching.  Edited  by 
John  Adams,  1918.  p.  195. 

Primary  Arithmetic.  1911. 

The  New  Methods  in  Arithmetic.  1921. 

Psychology  of  Arithmetic.  1921. 

Thorndlke  Arithmetics  (Three  books)  1918. 

Subtraction  by  the  Addition  Process.  Ele- 
mentary School  Journal,  Vol.  20,  Nov. 
1919.  pp.  203-207. 

Teaching  and  Testing  the  Teaching  of  Essen- 
tials. National  Educational  ass  oc^  s  tion. 
Report,  1913.  no.  56-57. 

Report  of  Committee  on  Elimination  of  Sub- 
ject Matter,  to  the  Iowa  State  Teachers 
Association,  Ames,  Iowa.   The  Sixteenth 
Yearbook  of  the  National  Sociot;,   or  the 
Study  of  Education. 


WILSON,  G.  M. 


- 


A  survey  of  the  .ocial  and  Business  U3e 
of  Arithmetic.   The  Sixl        jarbook  of 
the  National  Society  for  the  Study  of  Edu- 
cation. 1917.  pr>.  128-142. 

Course  of  Study  in  Mathematics.  Conner- 
ville,  Indiana  Public  School.  1911. 

A  Survey  of  the  Social  and  Busines'  Usage 
of  Arithmetic  -  Ph.D.  Thesi'g.  1919. 


WILSON,  II.  B.  and  G.M. 


Motivation  of  School  Vork.  1916.  pp. 
158-182:  370-451. 


WINCH,  V 


Accuracy  in  School  Children.   Does  Improve- 
ment in  Numeric?.!  Accuracy  Transfer  to 
Arithmetical  Reasoning?  Journal  of  Edu- 
cational Psychology.  Vol.  1.  1910.  pp. 
557-589;  Vol.  2.  1911.  pp.  262-271  and 
534-336. 


VISE,  C»RL  T. 


WOOD,  EKNEST  K 


Survey  of  Avithmetical  Problems  Arising 
In  Various  Occupations.   Elementary 
School  Journal,  Vol.  20.  Oct.  1919.  pp. 
118-136. 

Test  on  Efficiency  in  Arithmetic.  Ele- 
mentary School  Journal.  Vol.  17,  ^pp* 
446-453. 


V00DY,  C. 

yocdm,  a.  Duncan 

YOUNG,  J.  ...  A. 


Measurements  of  Some  of  the  Achievements 
in  Arithmetic.  School  and  Society.  Vol. 
4,  pp.  299-303. 

Culture,  Discipline  and  Democracy. 

The  Teaching  of  Mathematics  in  the  Ele- 
mentary and  Secondary  School.  1907. 


-1- 


APPENDIX . 

CKITICISM  AND  SUGGESTIVE  CHANGES  FOR  THORNDIKE'S  AHITMMETICS } 
Book  One.   Part  One.  Third  grade. 

Pa-e  20.  No's.  12,  13,  14  have  little  or  no  meaning  to 
8  year  old  children.   Same  is  true  of  No.  13  page  22. 

Eliminate  all  "counts"  except  from  0.  See  page335,  37, 
79,  80,  92,  118,  100  (first  10),  120. 

Pages  42,  43,  and  135.  " which  number  means  dol- 
lars etc.".   There  is  only  one  number  representing  dollars  and 
cents.  The  vord  number  should  be  replaced  by  part  or  figures . 

Page  44. —  "So  increase  the  7  to  17."  Impossible. 
"Increase"  should  be  eliminated  and  what  is  actually  done  (added) 
stated;  as,  add  10  to  7  etc. 

Page  46.  Eliminate  No's.  7,  8,  9,  and  10  as  they  are 
of  no  value  to  third  grade  pupils. 

Pages  65,  106,  107,  108  and  111  are  not  for  third  grade 

Pages  66,  70,  and  71.   "Write  3  In  the  tens  column." 
There  is  no  tens  column.   Better, —  "in  the  tens'  place." 

Pages  74  and  75  and  No's.  14  to  18  inclusive,  might  do 
for  fifth  or  sixth  :ri  c,  tit  not  for  third. 

Page  87.   "Write  4  over  the  0  of  30."  Write  5  over  the 
0  of  90. 

1.   Note  pagers,  note  3. 


-2- 

Pages  88,  89,  90  are  better  for  the  fourth  grade. 
Book  One.   Part  Two.  Fourth  grade. 

For  lower  grades,  especially,  the  form  of  the  fraction 
should  have  a  horizontal  line;  as,j  instead  of  1/3.  6t  vs .  6  1/3. 

Page  127.  No.  6,  Is  psychologically  incorrect.   Better 
divide  by  2,  3,  etc.  as  we  would  if  using  875. 

Page  133.  What  is  the  object?  Eliminate. 

Page  142,  143,  144,  145,  173  better  be  placed  in  Book 
Two  -  Part  Two. 

Page  150.  Why  such  problems  for  10  year  old  puoils? 
Eliminate. 

Page  159.   No  child  below  the  high  school  can  ansver 
No's.  5  and  6,  or  do  14  and  15,  or  cut  a  pie  into  fifths.  Eliminate 

Pare  166  No's.  5  and  8.   "Write  the  1  after  the  quotient 

."   1  over  the  divicor  is  a  part  of  the  quotient.   "Place  1 

over  the  divisor  as  a  part  of  the  quotient"  is  better. 

Page  177.  Change  directions  for  dividing  by  a  three 

figure  divisor.  Children  cannot  "think"  three  or  ""ore  finrures 

into  a  dividend. 

Pan;e  187.  No.  2  better  multiply  by  »  first  and  make  but 

o 

one  addition.   Page  191,  same  criticism. 

Why  pacre  199  with  Fourth  grade,  or  any  other?  Eliminate. 

All  the  work  of  adding  and  subtracting  mixed  numbers  and 
chan Ting  Improper  fractions  and  mixed  numbers  should  be  put  in 


-3- 

fifth  grade. 

Book  Two .   Part  One .  Fifth  grade . 

Page  10.  No's.  1,  2,  5,  and  6,  change  "Increase"  to 

add  1,  or  —  etc. 

8  i 

Page  24,  "Numbers  like  „  --",  change  to  "fractions  like 
1 
8  etc.  and  fractions  like  IT  etc. 

Page  32.  Eliminate  counts,  No's.  1,  2,  and  3. 

Page  72.   " which  numbers  mean  miles  and."  There 

Is  only  one  number  in  the  product.   Should  be  which  part  or  fig- 
ures mean  miles,  etc. 

Page  74.  Eliminate  No's.  17,  20,  23,  29,  and  33,  also 
Page  82  No's.  1,  2,  3,  4,  and  5. 

Page  75.  V.hy  not  state  where  to  put  the  decimal  point 
instead  of  requiring  the  teacher  to  tell  where  it  belon.cs?  The 
work  with  decimals,  pages  67  to  90  and  99  to  113  incltisive 
should  be  in  part  two,  for  sixth  grade,  also  the  work  with  com- 
pound denominate  numbers,  pages  91  to  97  inclusive. 

The  fifth  'rade  needs  more  work  with  fractions.   If  it 
be  taken  from  the  fourth  grade  work  and  put  into  Book  Two,  Part 
One,  the  work  for  both  grades  will  be  improved. 

In  multiplying  a  mixed  number,  or  by  a  mixed  number, 
the  fraction  should  be  used  first. 


Book  Two.   Part  Two.   Sixth  grade. 

Pa  e   137.  Eliminate  complex  fractions,  exceet  the  ali- 
quot parts  of  100,  or  1000. 

Page  142.   Eliminate  No^.  3  and  6.  The  answer  to  No. 
6  is  nothing. 

Pages  200  (Commission),  201  (Gains  and  Losses),  202 
(Fixing  Prices).  205  (Sharing  in  percents ) ,  206  (Interest)  and 
207,  should  he  put  into  Book  Three,  for  7th  grade.  The  same  is 
true  of  Mensuration,  pages  221  to  231. 

Pages  243,  111  (Commission).  244,  245,  246,  and  248  shoul 
be  placed  in  Book  Three  for  7th  grade. 

Taking  from  the  fifth  grade  and  giving  to  the  sixth,  and 
taking  from  the  sixth  and  giving  to  the  seventh,  as  indicated, 
will  make  both  divisions  of  rrork  within  bounds  of  what  can  be  done. 

Book  Three.   Part  una.  Seventh  grade. 

Page  2.   " —  adding  thousandths  — "  should  be  number  one. 

w  2.  H--  adding  hundredths  — "  should  be  number  two. 

"  2.   ■ —  addi-ig  ones ■  number  three. 

"   2.   ■ —  adding  tens  "  number  four. 

Page  6.  Eliminate  counts . 

Page  13.  Form  of  checking  (no.  17)  never  in  business. 
Page  15.  To  divide  by  or  multiply  by  a  "number".  What 
about  fractions? 


Page  16.   "Multiply  the  integer  and  numerator."  Sub- 
stitute by_  for  and. 

There  appears  but  little  new  work  for  the  seventh  grade 
The  percentage  vork,  and  the  mensuration  riven  for  the  sixth 
grade  is  needed  for  the  seventh. 

Book  Three.   Part  Two.  Eighth  grade. 

Work  for  the  eighth  grade  is  good  but  the  quantity  is 
out  of  proportion  to  that  given  for  the  seventh  grade. 


E  ubrm,y  of  rauwjj?" 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 

— ^\,,:^iii3  0MW — 


JUL   14  1952 


LD  21-95m^l 


,'50(2877sl6)476 


,  KoJth- 

y   of  ellmi> 
■ 
and  busineje   requirement 
of  arithrt-.  ' 


Spiers 

A  etud 


676f?23 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


BERKELEY,  CALIFORN/A 


